jmc

Let the coordinates of $T$ be $(x,y)$. Then, the midpoint of $\overline{PT}$ is $\left(\frac{x+5}{2},\frac{y+3}{2}\right)$. Since we know the coordinates of this point are those of point $Q$, $(-3,6)$, we have $(x+5)/2=-3$ and $(y+3)/2=6$. Solving these equations gives $x=-11$ and $y=9$, so point $T$ is at $\boxed{(-11,9)}$.