One of our mechanics at work had a broken bolt he was removing and he had a snap on tool. I was doubtful that it would work,however I watched him remove that bolt without a problem. Needless to say I bought one from snapon and have used it several times with no problems. Maybe I was lucky,but this is the first remover that has ever worked for me.
There is several different sizes of drill bits with a tool that is shaped like an allen wrench that you tap into the drilled hole of what ever you are trying to remove. Then there is an adapter that fits over the little allen wrench. Carefully use a wrench on the adapter and the broken bolt screws out. I always try to drill to the full depth of the bolt or screw that I am trying to remove. Maybe I have been lucky.
The mechanic told me that it was better because the force was through the bolt and not screwing into the bolt causing more outward pressure which made things tighter. His comments made sense to me.

Just what has worked for me. Not cheap to buy but worth it.
Cheers Ed