A standard example of Nakajima quiver varieties are type A Springer resolutions $\widetilde{\mathcal{N}} \to \mathcal{N}$. In the theory of Springer resolutions it is often beneficial to consider the full Grothendieck-Springer resolution, i.e. to work with the commutative diagram

$$\require{AMScd}\begin{CD} \widetilde{\mathcal{N}} @>>> \widetilde{\mathfrak{g}} \\ @VVV @VVV\\ \mathcal{N} @>>> \mathfrak{g} \end{CD}$$

Is there a similar diagram where the left side is replaced by $\mathfrak{M} \to \mathfrak{M_0}$ for an arbitrary quiver variety.

To be clear, I do not expect that the analogue of the Grothendieck-Springer resolution is necessarily a quiver variety itself, I merely ask whether to each (nice?) quiver variety one can associate a space that behaves similar to $\widetilde{\mathfrak{g}}$. I'm also intentionally vague about the word "similar", but for example a main feature of the above diagram is that it "lives over" $0 \hookrightarrow \mathfrak{h}//W \leftarrow \mathfrak{h}$.

Alternatively, why should I not expect such a thing to exist?