drilling through the stud is the easy part. just make sure it's dead center as much as possible and straight, obviously.
trying to drill out an extractor that is broken off is near impossible, the only way to get an extractor out is to either
FIND a drill bit EVEN HARDER than the extractor (good luck with that, not cheap) or break/fracture the extractor up even more and remove it in pieces using a hard tempered center punch and a hammer and picking the pieces out with a dental tool. been there done that.
now keep in mind, there should be a little dead-space in the head under the studs. when I drill my studs out in the past I make sure to drill completely through the stud but not into the head. this will relieve as much of the pressure on the threads as you can. I then use another drill bit that almost cuts the backside of the threads of the stud itself, almost creating a helicoil out of the broken stud. if you can drill a stud out in this method, you shouldn't even need an easy out to remove what's left, the right sized tap will remove what's left.