I agree interviews like this cannot be conducted in a closed session under the Open Meeting's Act. Having the mayor or perhaps a "hiring committee" (made up of the mayor, a council member and sometimes the retiring employee) are how most municipalities I've worked with have done it.
I also recommend, if you haven't already done so, to check your town's municipal ordinances to make sure there isn't an ordinance which directs hiring to be done in a certain way. I know a couple of the municipalities I work with have ordinances which specifically direct the mayor to do the interviews and hiring (although the hiring is typically subject to council approval) and I have one municipality that has an ordinance which directs the council to appoint someone to do the interviews and then make a recommendation to the full board.
Jamian J. Simmons
Smith, King & Simmons P.C.
P.O. Box 302
Gordon, NE 69343
(308) 282-0690