N.H. Rev. Stat. Ann. § 670:12

The commissioners shall make appointments to fill vacancies in the offices of village district clerk, moderator, and treasurer, and their own board. In the case of a vacancy in the office of commissioner with a 2- or 3-year term, the remaining commissioners shall make an appointment to fill said vacancy until the next annual meeting of the district, at which time a commissioner shall be elected to fill the unexpired term.