A circle of radius $2\sqrt{3}$ is centered at $O$. Square $OABC$ has side length $\sqrt{3}$. Sides $AB$ and $CB$ are extended past $B$ to meet the circle at $D$ and $E$, respectively. What is the area of the shaded region in the figure, which is bounded by $BD$, $BE$, and the minor arc connecting $D$ and $E$?