$\newcommand{\g}{\mathfrak g}$
Let $G$ be a reductive group and $U_q(\g)$ the associated quantum group. One can argue that the classical limit of $U_q(\g)$ is $G$ or $\g$, with some Poisson structure, but here's a picture I like better about the category of modules over those:

The basic observation is that a $G$-module is the same as a $G$-equivariant quasi-coherent sheaf on a point, i.e. a quasi-coherent sheaf on the quotient stack $BG=pt/G$. This stack has a 2-shifted symplectic structure, and one can define a Poisson sigma model which to a topological surface $S$ attaches 
$$QC(Map(S,BG)).$$
So this thing is a categorified 2-dimensional topological field theory, or equivalently the 2-dimensional part of a would be 4-dimensional field theory. Note that $Map(S,BG)$ is identified with the moduli stack of $G$-local systems on $S$ so this thing really is the classical limit of Chern-Simons theory. This is in accordance with the old idea that Chern-Simons/Reshetikhin-Turaev really is secretly a 4, not 3, dimensional TFT. 

Hence it is expected that Khovanov homology and its generalizations should come from a partially defined 5-dimensional TFT, which mathematically should be computed by a still conjectural braided monoidal 2-category structure on categorified quantum groups. Witten has a lot of ideas about the physical side of this story, which I honestly don't understand.

So I'm wondering if at least the classical part of this theory has a nice description similar to he one above, i.e. a nice geometrical construction from $G$ of a 2-dimensional TFT valued in (dg) 2-categories.