Good point! Is this the voice of experience talking?
Yes, at least when I was younger anyways. I'd like to say I have more control of my temper these days but more importantly, I have better tools now so a broken bolt isn't too big of a deal.
So, proceed with caution and don't be in a hurry. Use a decent center punch to mark the center of the screw head and start drilling with a small drillbit and tapping oil or WD-40. Work your way up one bit size at at time till you start to get into the threads. Sometimes you get lucky and the heat and oil are enough to break the bolt loose, other times you have to take it right up to the threads and then drive it out with a pick or chisel.
I've heard of guys using the tip of an engraving tool to work the bolt out as well but I haven't had much luck with that one myself.