  We generalize the topological recursion of Eynard-Orantin (2007) to the
family of spectral curves of Hitchin fibrations. A spectral curve in the
topological recursion, which is defined to be a complex plane curve, is
replaced with a generic curve in the cotangent bundle $T^*C$ of an arbitrary
smooth base curve $C$. We then prove that these spectral curves are
quantizable, using the new formalism. More precisely, we construct the
canonical generators of the formal $\hbar$-deformation family of $D$-modules
over an arbitrary projective algebraic curve $C$ of genus greater than $1$,
from the geometry of a prescribed family of smooth Hitchin spectral curves
associated with the $SL(2,\mathbb{C})$-character variety of the fundamental
group $\pi_1(C)$. We show that the semi-classical limit through the WKB
approximation of these $\hbar$-deformed $D$-modules recovers the initial family
of Hitchin spectral curves.
