If a bolt head is broken off because someone has tightened it too tight, you can usually get it out with an "easy-out" because when the head snaps off, the load on the bolt relaxes and the thread loosens up.
Exhaust bolts usually snap because they have become frozen in the hole and shear off when trying to remove them. This does nothing to loosen the bolt. If you try to remove it with an easy-out, you will most likely end up with a frozen bolt with a broken easy-out in the center. This solves all your problems because you can not drill an easy out, it is too hard. So now you can just junk the part you were working on and buy a new one. Or.....
If the bolt still prodrudes from the surface of the surrounding metal enough to get vise grips on it, apply some heat from a propane torch and liberal applications of liquid wrench or similar. Clamp the vise grips on the bolt and try to turn the bolt a very small amount in either direction. The bolt is flexible and will twist a small amount. By carefully twisting it back and forth, you have a chance of working the liquid wrench into the thereads until it breaks loose. If it is too difficult, let it soak for a day and try again.
If the bolt is broken off flush, or after the above method has failed and you have chewwed off the bolt end, center punch the bolt, flatten the end with a file if necessary and drill it very straight with a small drill, about 1/8".
This is the most important hole so have someone verify that the drill is perfectly straight and centered in the bolt.
Then select a drill somewhat smaller than the root diameter of the thread and re-drill the hole. You should now be able to see how close to the center of the hole you really are.
Select a drill which will just break through the thread root on the closest side and drill again. Often, when you are doing this, the remaining thread will break loose and you can twist it out with a pick or small screwdriver.
In any event, you will eventually be able to pick out the remnants of the bolt.
If the thread in the hole is destroyed, buy a "Heli-Coil" kit of the correct size from an auto parts and follow the directions.
When you finally replace the bolts, use anti-seize on the threads before re-assembly, you will love yourself next time you have to take it apart.