N.H. Rev. Stat. Ann. § 301:19

When a vacancy on the board of directors occurs other than by expiration of term, the remaining members of the board, by a majority vote, shall fill the vacancy, unless the bylaws provide for an election of directors by districts, and in such a case the board of directors shall immediately call a special meeting of the members, or stockholders, to fill the vacancy, in the district where it exists.