  We study spin Hurwitz numbers, which count ramified covers of the Riemann
sphere with a sign coming from a theta characteristic. These numbers are known
to be related to Gromov-Witten theory of K\"ahler surfaces and to
representation theory of the Sergeev group, and are generated by BKP
tau-functions. We use the latter interpretation to give polynomiality
properties of these numbers and we derive a spectral curve which we conjecture
computes spin Hurwitz numbers via a new type of topological recursion. We prove
that this conjectural topological recursion is equivalent to an ELSV-type
formula, expressing spin Hurwitz numbers in terms of the Chiodo class twisted
by the $2$-spin Witten class.
