Wipe Effect w/ closed shape

I have a stroked shape that I want to reveal with a simple wipe effect or wipe animation. But the default Wipe effect isn't quite what I want.

When I apply the standard Linear Wipe (Effects > Transition > Linear Wipe), the shape is revealed with the leading edge unstroked (the standard behavior for this effect).

I am desperately trying to figure out how to do this type of wipe, but with the shape "closed" throughout the progress of the wipe.

Any ideas are greatly appreciated.

I'm not necessarily looking for an easy way to do this: if you can think of a hard way, I'd like to hear it too.

Thanks!!!!