Yet worse example: take $U$ to be the complement of a pseudo-arc $P$ in the unit sphere. For any open disc $D$, such that $P \not\subset \bar D$ and $D\not\subset U$ we have $D\cap U$ is not connected.
Take a point in $P$ as the north pole and consider the stereographic projection of $U$ to the plane. For the obtained set $U'$ and any open topological disc $D$ such that $D\not\subset U'$ the set $D\cap U'$ is not connected.