N.H. Rev. Stat. Ann. § 541-B:5

A board member may disqualify himself relative to any matter before the board or if the board votes that any member has or may have a conflict of interest in any matter before the board, that member shall be disqualified to sit as a board member on that particular matter. In the event of any disqualification, the governor by the authority of RSA 21:33-a, the chief justice of the supreme court by the authority of RSA 541-B:3, II, the president of the senate or the speaker of the house of representatives, as appropriate, by the authority of RSA 541-B:3, III, shall appoint an interim member to the board to serve only as to that matter. The interim member shall have the same qualifications as the disqualified member.