Is there a way to change the widths of a compound path's strokes independently?

The stroke width tool in Illustrator features some really unpredictable behaviour when used on a compound path. When I change the width on one part of the compound path, Illustrator applies that width change to the other part as well.

Example:

This is a compound path: a circle with a smaller circular hole in it. I need it to be a compound path to have the hole. However, in the right picture I applied the Stroke width tool to the outer path's top anchor only, yet Illustrator goes ahead and applies the change to the inner circle as well.

Is there a way to change the widths of a compound path's strokes independently?