Broke the exhaust stud then the extractor what now?

I've been there, and I can feel your pain! Here's a couple of things I have had some luck with. Try to weld a screw or something onto the extractor and use it as a handle to wiggle it out. A TIG torch is the way to go if you have access to one, but one of those "stud gun" dent pullers might work if you have room to swing it.

Try slotting the end of the extractor with a cutoff wheel in a Dremel tool and see if you can turn it out with a screwdriver. Or just slot all the way through it. Since it is outboard of the port, you shouldn't hit water. The manifold should cover the incision when you are done, but don't slot into the port or you will have an exhaust leak.

I have also used a oxy-acetylene welding torch to draw the temper out of the extractor/drill bit/tap so it can be drilled.