N.H. Rev. Stat. Ann. § 128:1

The commissioner of the department of health and human services shall appoint as health officer for each town such person as the selectmen of the town recommend, and the commissioner of the department of health and human services shall issue to the health officer a certificate of appointment. The appointed officer shall provide the department with sufficient and complete contact information to be readily accessible as needed. Upon the expiration of the officer's term, or upon his or her removal, death, or resignation or the occurrence of a vacancy in the office for any reason, the selectmen shall recommend a replacement as soon as reasonably practicable; but if no recommendation is made within 15 days after notice, the commissioner may appoint a health officer without such recommendation.