Given a simple closed curve in the plane, there is a homeomorphism from the unit open disk to the interior of the curve. The homeomorphism can be taken conformal, this is the uniformisation theorem.

Is there a version of this result for closed curves that are not simple?

For example, given a $C^1$ closed curve $\gamma: S^1 \rightarrow {\bf R}^2$ with finitely many self-intersections, all of them transverse, is there a continuous map (maybe even conformal) from the unit open disk to the plane, such that the number of preimages of any point in ${\bf R}^2\backslash \gamma(S^1)$ is equal to the absolute value of the number of turns the curve makes around the point?