How do I break the selected UV map from its surrounding vertices?

You can do this by selecting the face and using Select > Select Split from the Image Editor header or the Y key, then drag to separate. This method doesn't require you changing the Sticky Selection Mode, but it does require that you separate at least one tri. This method does not work for separating a single point or edge.
Change the Sticky Selection Mode to Disabled in the bottom settings bar, then drag to separate the selection.
