N.H. Rev. Stat. Ann. § 127:5

The board shall appoint some discreet person, possessing the qualifications hereinafter specified, to be health officer for such district. If the district health officer shall not be elected within 60 days from the formation of any such district, or in case a vacancy in said office shall continue to exist for 60 days or more, such officer shall be appointed by the commissioner of the department of health and human services. Upon the appointment of a health officer under the provisions of this section, the terms of office of the health officers of the towns or cities forming such district shall terminate.