In N=2 Quantum Field Theories and Their BPS Quivers by Alim, Cecotti, Córdova, Espahbodi, Rastogi and Vafa the authors give a recipe which constructs from a pair $(C,\phi)$ consisting of a Riemann surface $C$ and a meromorphic quadratic differential $\phi$ on $C$ a (non-compact) Calabi-Yau 3-fold $X$.

Essentially, $X$ is cut out from the total space of the bundle $E=T^*C\oplus T^*C\oplus T^*C$ by the equation $uv=y^2-\phi(z)$ where $z$ is a local coordinate on $C$ and $(u,v,y)$ are coordinates on the fibre $E_z$. An important feature is that $X$ contains the **spectral curve** of $(C,\phi)$, i.e., the curve $\Sigma\subset T^*C$ which is a branched double cover of $C$ branched at the critical points of $\phi$.

From poking around physics literature that I don't understand, I get the sense that this construction is in some sense *post hoc* and the natural construction is to go the other way. That is, one should start with (some particular type of) CY $X$ and extract from it the pair $(C,\phi)$. Hence my question:

Is there a natural recipe which takes a CY $X$ and returns the pair $(C,\phi)$, inverse to the construction above?

Any insights or references to the literature would be greatly appreciated.