AGREEMENT

Exhibit 10.6

AGREEMENT

This Agreement (the “Agreement”) is entered into by and between Randolph Savings Bank (the “Bank”) and Kellie J. Lally (the “Employee”). The effectiveness of this Agreement is contingent upon the closing (the “Closing”) of the transactions contemplated by the Agreement and Plan of Merger among Randolph Bancorp, First Eastern Bankshares Corporation (the “Seller Company”), and Richard F. Kalagher, dated as of September 1, 2015 (the “Merger Agreement”). This Agreement shall be effective upon the date of the Closing (the “Effective Date”).

1. Purpose. The Employee has been employed in a senior capacity by First Federal Savings Bank of Boston (the “Seller Subsidiary”), which is a wholly-owned subsidiary of the Seller Company. The Board of Directors of the Bank (the “Board”) believes that the Employee has the skills, knowledge and experience to be an important and effective contributor to the Bank’s success. The Board seeks to encourage the Employee’s employment with and commitment to the Bank.

2. Certain Employment Terms. The Employee’s employment with the Bank shall commence on the Effective Date. The following terms shall apply to the Employee’s compensation and employment responsibilities during the Employee’s employment with the Bank.

(a) Position and Responsibilities. The Bank shall employ the Employee in a management-level position or positions appropriate for an individual of the Employee’s skills, knowledge and experience with responsibilities consistent with such skills, knowledge and experience.

(b) Salary. The Bank shall pay the Employee a base salary at an annual rate equal to the Employee’s annual base salary rate as an employee of the Seller Subsidiary as of the Effective Date.

(c) Bonus. The Employee shall be eligible to receive an annual bonus pursuant to and subject to the terms of any annual short-term incentive plan that is in place for other members of the Bank’s management team.

(d) Benefits. The Bank shall provide benefits to the Employee in accordance with Sections 7.6(a) – (f) of the Merger Agreement, which Sections are incorporated herein by reference.

3. At-Will Employment. The Bank’s employment of the Employee shall be on an at-will basis; provided that (i) the Employee shall give at least 30 days’ notice of the Employee’s resignation from employment with the Bank; and (ii) the at-will nature of the Employee’s employment shall not limit the Employee’s rights under Section 4 (Severance Pay). If the Employee gives advance notice of the Employee’s resignation, the Bank may accelerate the termination of the Employee’s employment by paying the Employee’s salary at the Employee’s base salary rate for any period of acceleration. Such accelerated resignation shall not affect the treatment of the Employee’s termination as a resignation.

4. Severance Pay. If prior to the five year anniversary of the Effective Date, the Employee’s employment is terminated by the Bank for any reason other than for Cause, Disability or death, or if the Employee resigns for Good Reason, then subject to the Employee signing a separation agreement to be proposed by the Bank, which shall be substantially in the form attached hereto as Exhibit A, amended as reasonably necessary based on applicable law (the “Separation Agreement and Release”) and the Separation Agreement and Release becoming irrevocable, all within 45 days after the date of termination, the Bank shall pay to the Employee a severance amount equal to the Employee’s annual base salary as of the date of termination (the “Severance Pay”). The Severance Pay shall be paid in a lump sum within 45 days of the date of termination; provided, however, that if the 45-day period begins in one calendar year and ends in a second calendar year, the Severance Pay shall be paid in the second calendar year no later than the last day of such 45-day period.

5. Definitions. For purposes hereof, the following terms shall have the meanings set forth below:

(a) “Cause” shall mean a termination of the Employee’s employment which is a result of: (i) actions and/or omissions by the Employee satisfying the elements of (A) a felony or (B) a misdemeanor involving dishonesty, the taking of property or moral turpitude; or (ii) willful disclosure of material trade secrets or other material confidential information related to the business of the Bank or any of its subsidiaries or affiliates; or (iii) willful and continued failure substantially to perform the Employee’s duties with the Bank (other than any such failure resulting from the Employee’s incapacity due to physical or mental illness) after a written demand for substantial performance is delivered to the Employee by the Board, which demand identifies the specific actions which the Board believes constitute willful and continued failure substantially to perform the Employee’s duties, and which performance is not substantially corrected by the Employee within fourteen (14) days of receipt of such demand; (iv) willful and knowing participation in releasing false or materially misleading financial statements; (v) dishonesty to the Board regarding any material matter; or (vi) failure to cooperate with a bona fide internal investigation or an investigation by regulatory or law enforcement authorities, after being instructed by the Bank to cooperate, or the Employee’s willful destruction or failure to preserve documents or other materials known to be relevant to such investigation or the inducement of others to fail to cooperate or to produce documents or other materials in connection with such investigation.

(b) “Disability” shall mean that if, as a result of the Employee’s incapacity due to physical or mental illness, the Employee shall have been absent from the Employee’s duties to the Bank on a full-time basis for 120 business days in the aggregate in any 12 month period or is reasonably expected based upon medical information to be absent for such period. If any question shall arise as to whether during any period the Employee has a Disability, the Employee may, and at the request of the Bank shall, submit to the Bank a certification in reasonable detail by a physician selected by the Bank to whom the Employee has no reasonable objection as to whether the Employee is incapacitated and how long any incapacity is expected to continue. Such certification shall, for the purposes of this Agreement, be conclusive of the issue. The Employee shall cooperate with any reasonable request of the physician in connection with such certification. If such question shall arise and the Employee fails to submit such certification, the Bank’s determination of such issue shall be binding on the Employee.

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(c) “Good Reason” shall mean the occurrence of any of the following events (each, a “Good Reason Condition”) followed by the Good Reason Process, as defined below: (i) a material adverse change in the Employee’s position and/or responsibilities such that the Employee no longer holds a position with responsibilities appropriate for an individual of the Employee’s skills, knowledge and experience; or (ii) a material reduction in the Employee’s annual base salary as in effect on the date hereof or as the same may be increased from time to time hereafter, except for across-the-board reductions of annual base salary similarly affecting all executive officers of the Bank; or (iii) the removal of the Employee from eligibility for any short-term incentive plan for members of the Bank’s management team for which other members of the Bank’s management team remain eligible or the application to the Employee of adverse changes in such plan’s terms that are not applied generally to other management-level participants; provided that “adverse changes in such plan’s terms” shall not be construed to include changes in bonus targets or good faith determinations of bonus amounts; or (iv) the relocation of the Bank’s offices at which the Employee is principally employed (the “Current Offices”) to any other location more than 25 miles of driving distance from the Current Offices, or the requirement by the Bank for the Employee to be based more than 25 miles of driving distance away from the Current Offices, except for required travel on the Bank’s business to an extent substantially consistent with the Employee’s business travel obligations. Notwithstanding the foregoing, a suspension of any or all of Employee’s duties or responsibilities by the Bank during an investigation that is initiated pursuant to a direction by the Board shall not constitute the occurrence of a material adverse change for purposes of clause (i); provided that Employee’s base salary and fringe benefit entitlements continue during the period of such suspension. The “Good Reason Process” shall mean that (i) the Employee reasonably determines in good faith that a Good Reason Condition has occurred; (ii) the Employee gives the Bank written notice of the first occurrence of the Good Reason Condition within 30 days of the first occurrence of such condition; (C) the Employee cooperates in good faith with the Bank’s efforts for a period of not less than 30 days following such notice (the “Cure Period”) to remedy the Good Reason Condition; (D) notwithstanding such efforts, the Good Reason Condition continues to exist; and (E) the Employee terminates employment by giving written notice no later than 30 days after the expiration of the Cure Period. If the Bank cures the Good Reason Condition during the Cure Period, Good Reason shall be deemed not to have occurred.

6. Withholding. All payments made by the Bank under this Agreement shall be net of any tax or other amounts required to be withheld by the Bank under applicable law.

7. No Mitigation. The Bank agrees that, if the Employee’s employment by the Bank is terminated prior to the five year anniversary of the Effective Date, the Employee is not required to seek other employment or to attempt in any way to reduce the Severance Pay. Further, the Severance Pay shall not be reduced by any compensation earned by the Employee as the result of employment by another employer, by retirement benefits, by offset against any amount claimed to be owed by the Employee to the Bank or otherwise, except as otherwise provided by this Agreement.

8. Confidential Information and Bank Property.

(a) Confidential Information. As used in this Agreement, “Confidential Information” means information belonging to the Bank which is of value to the Bank in the

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course of conducting its business and the disclosure of which could result in a competitive or other disadvantage to the Bank. Confidential Information includes, without limitation, financial information, reports, and forecasts; inventions, improvements and other intellectual property; trade secrets; know-how; software; market or sales information or plans; customer lists; and business plans, prospects and opportunities (such as possible acquisitions or dispositions of businesses or facilities) which have been discussed or considered by the management of the Bank. Confidential Information includes information developed by the Employee in the course of the Employee’s employment by the Seller Subsidiary or the Bank, as well as other information to which the Employee may have access in connection with the Employee’s employment. Confidential Information also includes the confidential information of others with which the Bank has a business relationship. Notwithstanding the foregoing, Confidential Information does not include information in the public domain, unless due to breach of the Employee’s duties under Section 8(b).

(b) Confidentiality. The Employee understands and agrees that the Employee’s employment creates a relationship of confidence and trust between the Employee and the Bank with respect to all Confidential Information. At all times, both during the Employee’s employment with the Bank and after its termination, the Employee will keep in confidence and trust all such Confidential Information, and will not use or disclose any such Confidential Information without the written consent of the Bank, except as may be necessary in the ordinary course of performing the Employee’s duties to the Bank or as otherwise may be required by law.

(c) Documents, Records, etc. All documents, records, data, apparatus, equipment and other physical property, whether or not pertaining to Confidential Information, which are furnished to the Employee by the Bank or any affiliate or are produced by the Employee in connection with the Employee’s employment will be and remain the sole property of the Bank. The Employee will return to the Bank all such materials and property as and when requested by the Bank. In any event, the Employee will return all such materials and property promptly following termination of the Employee’s employment for any reason other than death. The Employee will not retain with the Employee any such material or property or any copies thereof after such termination.

(d) Injunction. The Employee agrees that it would be difficult to measure any damages caused to the Bank which might result from any breach by the Employee of the promises set forth in this Section 8, and that in any event money damages would be an inadequate remedy for any such breach. Accordingly, subject to Section 9 of this Agreement, the Employee agrees that if the Employee breaches, or proposes to breach, any portion of this Section 8, the Bank shall be entitled, in addition to all other remedies that it may have, to an injunction or other appropriate equitable relief to restrain any such breach.

9. Arbitration of Disputes. Any controversy or claim arising out of or relating to this Agreement or the breach thereof or otherwise relating to the Employee’s employment with the Bank or its termination, including without limitation statutory claims (including without limitation statutory discrimination or wage claims) and common law contract and tort claims shall be settled exclusively by arbitration in accordance with the laws of the Commonwealth of Massachusetts and the Employment Arbitration Rules of the American Arbitration Association.

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Such arbitration shall be conducted in the City of Boston. In the event that any person or entity other than the Employee or the Bank may be a party with regard to any such controversy or claim, such controversy or claim shall be submitted to arbitration subject to such other person or entity’s agreement. Judgment upon the award rendered by the arbitrator may be entered in any court having jurisdiction thereof. Notwithstanding the foregoing, this Section 9 shall not preclude either party from pursuing a court action for the sole purpose of obtaining a temporary restraining order or a preliminary injunction in circumstances in which such relief is appropriate; provided that any other relief shall be pursued through an arbitration proceeding pursuant to this Section 9.

10. Waiver. No waiver of any provision hereof shall be effective unless made in writing and signed by the waiving party. The failure of either party to require the performance of any term or obligation of this Agreement, or the waiver by either party of any breach of this Agreement, shall not prevent any subsequent enforcement of such term or obligation or be deemed a waiver of any subsequent breach.

11. Notices. Any notices, requests, demands and other communications provided for by this Agreement shall be sufficient if in writing and delivered in person or sent by registered or certified mail, postage prepaid, to the Employee at the last address the Employee has filed in writing with the Bank, or to the Bank at its main office, attention of the Board.

12. Effect on Other Plans. The Employee hereby waives any rights to any severance benefits under any current or future Bank severance pay plan or program with respect to any termination of employment that renders the Employee eligible for Severance Pay pursuant to this Agreement. Subject to such waiver, nothing else in this Agreement shall otherwise limit the rights of the Employee under the Bank’s benefit plans, programs or policies An election by the Employee to resign for Good Reason under the provisions of this Agreement shall not be deemed a voluntary termination of employment by the Employee for the purpose of interpreting the provisions of any of the Bank’s benefit plans, programs or policies.

13. Entire Agreement. This Agreement constitutes the entire agreement between the parties with respect to the subject matter hereof and supersedes in all respects all prior agreements between the parties concerning such subject matter.

14. Amendment. This Agreement may be amended or modified only by a written instrument signed by the Employee and by a duly authorized representative of the Bank.

15. Governing Law. This contract shall be construed under and be governed in all respects by the laws of the Commonwealth of Massachusetts, without giving effect to such state’s conflicts of laws principles.

16. Obligations of Successors. The Bank shall require any successor (whether direct or indirect, by purchase, merger, consolidation or otherwise) to all or substantially all of the business or assets of the Bank to expressly assume and agree to perform this Agreement in the same manner and to the same extent that the Bank would be required to perform if no such succession had taken place.

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17. Additional Limitation. Anything in this Agreement to the contrary notwithstanding, in the event that any compensation, payment or distribution by the Bank to or for the benefit of the Employee, whether paid or payable or distributed or distributable pursuant to the terms of this Agreement or otherwise (the “Severance Payments”), would be subject to the excise tax imposed by Section 4999 of the Internal Revenue Code of 1986, as amended (the “Code”), then the benefits payable under this Agreement shall be reduced (but not below zero) to the extent necessary so that the maximum Severance Payments shall not exceed the Threshold Amount, as defined below. To the extent that there is more than one method of reducing the payments to bring them within the Threshold Amount, the Employee shall determine which method shall be followed; provided that if the Employee fails to make such determination within 15 business days after the Bank has sent the Employee written notice of the need for such reduction, the Bank may determine the amount of such reduction in its sole discretion.

The “Threshold Amount” shall mean three times the Employee’s “base amount” within the meaning of Section 280G(b)(3) of the Code and the regulations promulgated thereunder less one dollar ($1.00); and “Excise Tax” shall mean the excise tax imposed by Section 4999 of the Code, and any interest or penalties incurred by the Employee with respect to such excise tax.

18. Section 409A.

(a) Anything in this Agreement to the contrary notwithstanding, if at the time of the Employee’s “separation from service” within the meaning of Section 409A of the Code, the Bank determines that the Employee is a “specified employee” within the meaning of Section 409A(a)(2)(B)(i) of the Code, then to the extent any payment or benefit that the Employee becomes entitled to under this Agreement would be considered deferred compensation subject to the 20 percent additional tax imposed pursuant to Section 409A(a) of the Code as a result of the application of Section 409A(a)(2)(B)(i) of the Code, such payment shall not be payable and such benefit shall not be provided until the date that is the earlier of (A) six months and one day after the Employee’s separation from service, or (B) the Employee’s death; provided, however, that in the case of benefits, the Employee may elect to pay for the costs of such benefits during such delay period in exchange for reimbursement of such costs after the end of the delay period. Any such delayed cash payment shall earn interest at an annual rate equal to the applicable federal short-term rate published by the Internal Revenue Service for the month in which the date of separation from service occurs, from such date of separation from service until the payment.

(b) The parties intend that this Agreement will be administered in accordance with Section 409A of the Code. To the extent that any provision of this Agreement is ambiguous as to its compliance with Section 409A of the Code, the provision shall be read in such a manner so that all payments hereunder comply with Section 409A of the Code. The parties agree that this Agreement may be amended, as reasonably requested by either party, and as may be necessary to fully comply with Section 409A of the Code and all related rules and regulations in order to preserve the payments and benefits provided hereunder without additional cost to either party.

(c) To the extent that any payment or benefit described in this Agreement constitutes “non-qualified deferred compensation” under Section 409A of the Code, and to the extent that such payment or benefit is payable upon the Employee’s termination of employment, then such payments or benefits shall be payable only upon the Employee’s “separation from service.” The determination of whether and when a separation from service has occurred shall be made in accordance with the presumptions set forth in Treasury Regulation Section 1.409A-1(h).

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(d) The Bank makes no representation or warranty and shall have no liability to the Employee or any other person if any provisions of this Agreement are determined to constitute deferred compensation subject to Section 409A of the Code but do not satisfy an exemption from, or the conditions of, such Section.

[Remainder of Page Intentionally Left Blank]

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IN WITNESS WHEREOF, this Agreement has been executed as a sealed instrument by the Bank by its duly authorized officer, and by the Employee, as of the Effective Date.

[Signature page to Agreement with Kellie J. Lally]

EXHIBIT A

SEPARATION AGREEMENT AND RELEASE

I enter into this Separation Agreement and Release (the “Release”) pursuant to Section 4 of the Agreement between Randolph Savings Bank (the “Bank”) and me (the “Agreement”), which Agreement was effective as of the closing of the transactions contemplated by the Agreement and Plan of Merger between Randolph Bancorp and First Eastern Bankshares Corporation (the “Seller Company”), dated as of September 1, 2015 (the “Merger Agreement”). I acknowledge my timely execution and return and my non-revocation of this Release are conditions to the payment of the Severance Pay in Section 4 of the Agreement. I therefore agree to the following terms:

1. Release of Claims. I voluntarily release and forever discharge the Bank, its affiliated and related entities, its predecessors, successors and assigns, its employee benefit plans and fiduciaries of such plans, and the current and former officers, directors, shareholders, employees, attorneys, accountants and agents of any and all of the foregoing in their official and personal capacities (collectively referred to as the “Releasees”) generally from all claims, demands, debts, damages and liabilities of every name and nature, known or unknown (“Claims”) that, as of the date when I sign this Release, I have, ever had, now claim to have or ever claimed to have had against any or all of the Releasees. This includes, without limitation, the release of all Claims:

provided, however, that this release shall not affect my rights under the Bank’s Section 401(k) plan, any other “employee benefit plan” as defined in Section 3(3) of the Employee Retirement Income Security Act, 29 U.S.C. § 1002(3), my continuing rights under the Agreement (including the right to payment of any bonus for which an award has been determined but has not been paid during the term of employment), any statutory right to earned but unpaid wages, including

vacation pay, statutory or common law rights of indemnification or defense for claims against me based on my status and conduct as an officer of the Bank under any applicable insurance policy, contracts, governing documents or bylaws. In addition, nothing in this release shall affect my rights arising from any relationship that I may have with the Bank or any affiliated or related entity as a customer or a client. Furthermore, nothing in this release shall affect my rights to pursue Claims against individuals based on actions taken in their personal capacities that are unrelated in any way to my employment with the Bank or its termination.

I agree that I shall not seek or accept damages of any nature, other equitable or legal remedies for my own benefit, attorney’s fees, or costs from any of the Releasees with respect to any Claim released by this Release. I represent that I have not assigned to any third party and I have not filed with any agency or court any Claim released by this Release.

2. Ongoing Obligations. I reaffirm my ongoing obligations under the Employment Agreement, including without limitation my obligations under Section 8 (“Confidential Information and Bank Property”).

3. Litigation and Regulatory Cooperation. I further agree that for a period of two years from the date of the termination of my employment with the Bank (the “Cooperation Period”), I shall cooperate fully with the Bank in the defense or prosecution of any claims or actions now in existence or which may be brought in the future against or on behalf of the Bank which relate to events or occurrences that transpired while I was employed by the Bank. My full cooperation in connection with such claims or actions shall include, but not be limited to, being available to meet with counsel to prepare for discovery or trial and to act as a witness on behalf of the Bank at mutually convenient times. During the Cooperation Period, I shall also cooperate fully with the Bank in connection with any investigation or review of any federal, state or local regulatory authority as any such investigation or review relates to events or occurrences that transpired while I was employed by the Bank. Any cooperation pursuant to this paragraph is subject to the Bank’s commitment to reimburse me for any reasonable out-of-pocket expenses incurred in connection with my performance of obligations pursuant to this paragraph, based on the procedures for reimbursement applicable to employee business expense reimbursement under the Bank’s policies. In addition, if the Bank does not offer to provide me with legal services through the Bank’s counsel in connection with such cooperation, either due to such counsel’s determination that joint representation cannot be provided or for any other reason, I may condition my cooperation on the Bank’s agreement to reimburse me for any reasonable attorney’s fees that I incur in the providing any cooperation services pursuant to this paragraph.

4. No Assignment. I represent that I have not assigned to any other person or entity any Claims against any Releasee.

5. Right to Consider and Revoke Release. I acknowledge that I have been given the opportunity to consider this Release for a period ending twenty-one (21) days after the date when it was proposed to me. In the event that I executed this Release within less than twenty-one (21) days after such date, I acknowledge that such decision was entirely voluntary and that I had the opportunity to consider this Release until the end of the twenty-one (21) day period. To accept this Release, I shall deliver a signed Release to the Bank’s most senior Human Resources officer within such twenty-one (21) day period. For a period of seven (7) days from the date when the I

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execute this Release (the “Revocation Period”), I shall retain the right to revoke this Release by written notice that is received by such Human Resources officer on or before the last day of the Revocation Period. This Release shall take effect only if it is executed within the twenty-one (21) day period as set forth above and if it is not revoked pursuant to the preceding sentence. If those conditions are satisfied, this Release shall become effective and enforceable on the date immediately following the last day of the Revocation Period (the “Effective Date”).

6. Other Terms.

(a) Legal Representation; Review of Release. I acknowledge that I have been advised to discuss all aspects of this Release with my attorney, that I have carefully read and fully understand all of the provisions of this Release and that I am voluntarily entering into this Release.

(b) Binding Nature of Release. This Release shall be binding upon me and upon my heirs, administrators, representatives and executors.

(c) Amendment. This Release may be amended only upon a written agreement executed by the Bank and me.

(d) Severability. In the event that at any future time it is determined by an arbitrator or court of competent jurisdiction that any covenant, clause, provision or term of this Release is illegal, invalid or unenforceable, the remaining provisions and terms of this Release shall not be affected thereby and the illegal, invalid or unenforceable term or provision shall be severed from the remainder of this Release. In the event of such severance, the remaining covenants shall be binding and enforceable.

(e) Governing Law and Interpretation. This Release shall be deemed to be made and entered into in the Commonwealth of Massachusetts, and shall in all respects be interpreted, enforced and governed under the laws of the Commonwealth of Massachusetts, without giving effect to the conflict of laws provisions of Massachusetts law. The language of all parts of this Release shall in all cases be construed as a whole, according to its fair meaning, and not strictly for or against the Bank or me.

(f) Entire Agreement; Absence of Reliance. I acknowledge that I am not relying on any promises or representations by the Bank or any of its agents, representatives or attorneys regarding any subject matter addressed in this Release.

So agreed.

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