N.H. Rev. Stat. Ann. § 329-B:8

The governor and council may remove any member of the board for misconduct, incompetence, neglect of duty, or other sufficient cause after the member has been given a written statement of the charges and an opportunity to be heard regarding such charges. Any vacancy in the membership of the board occurring otherwise than by expiration of a member's term shall be promptly filled for the unexpired term, after which the replacement board member shall be eligible for one additional consecutive term if reappointed. The replacement board member shall be of the same qualification and appointed by the governor and council.