One Word For This.......Torture!
I do have a question about how they broke. Did they break on their own, or did they break when trying to remove them? If they broke on their own, it is remotely possible that the studs are sitting inside loose. Try drilling a pilot hole in the center and then use an tiny easy-out tool. If it doesn't unscrew easily with the tool, don't keep torquing it or the tool will break and then you have a bigger problem.
This is an easy-out tool.
Years ago I used to turn Pontiac Fiero-GTs which often suffered from self-breaking manifold bolts which always sheared off clean as your pictures shows....never leaving the stud exposed to work with. I got lucky a few times and was able to work out the stud with a fine-tip jewelers screwdriver. Once with an eraser-headed pencil.
Gently tap the stud with a punch to break it loose from the manifold threads. That sometimes works on stuck studs for my suggestions to be effective. One more thing, make sure the exposed opening is clean of rust. I see rust where that first thread starts. Get that real clean for the threads to get past the opening. A Dremel wire wheel may work but might not be aggressive enough.
One final suggestion would be to use a reverse-cut drill bit when drilling the pilot hole. Sometimes the stud starts unscrewing itself right out while drilling.
