Having had to do a fair amount of this both for profit and for "pleasure", the most recent being 4 of 5 broken off flush in a turbo housing, I would like to offer a few suggestions. The first is of course"don't break them"! Before you apply lots of torque you should be applying penetrating oil, heat, impact etc. I usually apply a wussy level of torque and then penetrating oil, hammer to the bolt head, heat to the bolt/stud followed by more oil. Impact wrenches seem to be better than a pure twist.
After breaking things off I check to see if the break is above the surface, flush, or below. Also if the mating hole is concentric with the broken stub. If it is I will make a temporary guide bushing on a lathe that will guide a drill by the existing hole in the mating part. This presupposes that you have left the part in place rather than just breaking everything off to get it loose! For unthreaded stubs I fabricate a stepped bushing that fits over the stub to guide the drill while threaded stub get a coupling nut with a threaded guide inserted. If things are broken off well inside the hole this threaded guide can sometimes be screwed into the hole. For the flush break you must eyeball the center as best possible and center punch. Recheck for centrality and repunch as required, angling the punch to move the dot. Then drill a shallow dimple with a small drill. Rechecking will show which way the dimple is off. Adding more punches on the thick side will move the hole slightly on further drilling and marking the dimple with a chisel is even better. Slanting the drill is less effective.
The object of all of the above is to end up with a small hole dead center in the broken section. Once you have this then you can enlarge it to take the proper size of Ezy-Out but don't be too brave or you can lose a LOT of ground! I prefer to just keep drilling the hole larger and larger until the threads show. Keep using the penetrating oil as once you can get it behind the stub it is much more effective. While I do not have a set of LH drills I certainly should. Once you are at the tap drill size the proper size tap will clear out the rest of the threads and clean upthe hole