Closest pair of points between two sets, in 2D
Yes, this can be $O(n \log n)$ time.  Build a Voronoi diagram for $T$.  Then, for each point $s \in S$, find which cell of the Voronoi diagram it is contained in.  The center of that cell is the point $t \in T$ that is closest to $s$.

You can build a Voronoi diagram in $O(n \log n)$ time, and each query (find the cell containing $s$) can be done in $O(\log n)$ time, so the total running time is $O(n \log n)$ time.