  We present a quantization of an isomorphism of Mirkovi\'c and Vybornov which
relates the intersection of a Slodowy slice and a nilpotent orbit closure in
$\mathfrak{gl}_N$ , to a slice between spherical Schubert varieties in the
affine Grassmannian of $PGL_n$ (with weights encoded by the Jordan types of the
nilpotent orbits). A quantization of the former variety is provided by a
parabolic W-algebra and of the latter by a truncated shifted Yangian. Building
on earlier work of Brundan and Kleshchev, we define an explicit isomorphism
between these non-commutative algebras, and show that its classical limit is a
variation of the original isomorphism of Mirkovi\'c and Vybornov. As a
corollary, we deduce that the W-algebra is free as a left (or right) module
over its Gelfand-Tsetlin subalgebra, as conjectured by Futorny, Molev, and
Ovsienko.
