Finding vertices of a maximum clique in polynomial time
You first give it $(G,k)$, and if it says yes, you do the following.  You pick an arbitrary vertex $v$ and call your oracle with $(G-v,k)$.  If the oracle says yes, throw away $v$ and forget about it.  If it says no, then $v$ is in every clique of size at least $k$.  Output $v$ and decrease $k$ by one and lather, rinse, repeat.