[theastronaut]

A carbide burr on an air grinder at high rpm will work wonders compared to a drill. Drills will need to be harder than the bolt, and drills that hard will be more likely to chip since they're more brittle. One chip and they stop cutting.

If its not recessed that deeply then the best way to get a broken bolt out is to build up tack welds on the shank until its flush, then weld a washer to the tacks, then weld a nut to the washer and back the whole thing out with a wrench or socket.