My post didn;t show up, so I'm going to try again.
There are a number of screw extractors here:
Woodworker's Supply
Although my method might revolve around a Dremel tool and a pair of needle nose vise grips.
The Dremel with a dental like engraver will allow you to make a slot. Once the screw clears flush, it can be flattened around the edges and removed with the vise grips.
If the head has to be ground off, then you may have to use the 1/4" to 1/2" screw extractor/hole saw to remove it. The hole is then patched with a glued in dowell.
I have an issue with TRYING to drill the screw out with a conventional drill. My feeling, is that if your not carefull, the drill will slip and damage the cabinet.
I have had the most success by slotting and using a conventional bit to remove. Without a flat surface and mark it's difficult to drill without the drill wandering.
Good luck.