  We prove that the topological recursion formalism can be used to compute the
WKB expansion of solutions of second order differential operators obtained by
quantization of any hyper-elliptic curve. We express this quantum curve in
terms of spectral Darboux coordinates on the moduli space of meromorphic
$\mathfrak{sl}_2$-connections on $\mathbb{P}^1$ and argue that the topological
recursion produces a $2g$-parameter family of associated tau functions, where
$2g$ is the dimension of the moduli space considered. We apply this procedure
to the 6 Painlev\'e equations which correspond to $g=1$ and consider a $g=2$
example.
