Does the intersection of VC and CLIQUE belong to NPC?
The graphs with clique size at least $k$ and VC at most $k$ have a particular
structure: They can be partitioned into three sets, $C$, $I$ and a singleton
$\{s\}$; $G[C]$ is a clique, $G[I]$ is an independent set and $s$ is just a
vertex.  $G[C \cup I]$ is what's called a split
graph.

Proof: If a graph $G$ contains a clique of size $k$, then that clique uses up
(at least) $k-1$ of your vertex cover budget.  Let $G,k$ be a yes instance and
$C$ any $k$-clique.  Observe that $G - C$ is a star with possibly isolated
vertices.

There are two cases: $G-C$ has edges, and $G-C$ does not have edges.  The latter
case is trivial, since we can let $C$ be the solution.  In the former case,
$G-C$ is the star graph with isolates.  Let $s$ be the vertex with non-zero
degree.

Both these cases lead to polynomial recognition algorithm.  The one in which
$G-C$ is an independent set, is the case where $G$ is a split graph.  The other
case has one vertex $s$ for which $G-s$ is a split graph.  Guess $s$, and you're
back at case 1.

As xskxzr pointed out, there is not necessarily a unique "split partition", that is, a partitioning of a split graph into the vertex sets $C$ and $I$, however, there are at most $n$ such partitions, and you can try all.  This leads in the end to an $O(n^3m)$ algorithm for VC∩C, which I'm sure can be improved.