  Double Hurwitz numbers enumerate branched covers of $\mathbb{CP}^1$ with
prescribed ramification over two points and simple ramification elsewhere. In
contrast to the single case, their underlying geometry is not well understood.
In previous work by the second- and third-named authors, the double Hurwitz
numbers were conjectured to satisfy a polynomiality structure and to be
governed by the topological recursion, analogous to existing results concerning
single Hurwitz numbers. In this paper, we resolve these conjectures by a
careful analysis of the semi-infinite wedge representation for double Hurwitz
numbers, by pushing further methods previously used for other Hurwitz problems.
We deduce a preliminary version of an ELSV-like formula for double Hurwitz
numbers, by deforming the Johnson-Pandharipande-Tseng formula for orbifold
Hurwitz numbers and using properties of the topological recursion under
variation of spectral curves. In the course of this analysis, we unveil certain
vanishing properties of the Chiodo classes.
