UPDATE:

So I drilled the crap out of the screw heads (I didn't mention earlier that this happened to two screws, one on each shoe... the right screw on both... coincidence?). Worked like a charm on the first shoe. Popped the cleat off with a screwdriver and then used a pair of needlenose pliers to extract the leftover screw shaft.

The second shoe is still in progress. I got the cleat off, but cannot get the screw out. I can't get enough purchase on the remaining part sticking out of the hole it's in and it's in there pretty damn tight.

Does anyone in NYC have a dremmel I can borrow for about 5 minutes? I think i'm gonna have to carve in a notch and use a flathead here...

Man. what a pain in the ass.

PS - I did use grease before I put the screws in btw. it's dry and brittle. it's been along time, with heavy use. it just dried out. maybe you have to regrease those threads every few months or something?