It's one of the worst kinds of feelings to experience while working on an exhaust system. You're installing or removing an exhaust stud or nut and you hear that sickening snapping sound that indicates the stud or bolt has broken. Now you have to completely uninstall the manifold or header and remove the broken stud or bolt. If the point of the break is at least 1/4 inch outside the head, you're lucky because you can use a stud extractor. If not, you need to drill it out and use a screw extractor.
Remove the manifold or header by removing the remainder of the nuts or bolts. Use a penetrating lubricant to ease removal.
Grasp the body of the extractor, with the jaws towards the head and twist the base clockwise to open the jaws and slide the broken stud end into the extractor.
Rotate the extractor counterclockwise with the wrench and remove the broken stud or bolt. This is also the preferred method of stud removal when you're replacing your current exhaust studs.
Place the point of the centre punch in the middle of the broken stud and strike it sharply to create a dimple to accept the drill bit. If the end of the stud or bolt sticks out any, flatten it with the file or grinding wheel to make this easier.
Clamp the drill bit in the drill, switch it to counterclockwise rotation and hold the tip of the bit in the dimple created in Step 1. Activate the drill, and drill the exact centre of the broken stud or bolt at least 1/8 inch deep, applying cutting or drilling fluid liberally. The fluid cools the drill bit tip and stud, and helps the bit cut more efficiently. Remove the drill bit from the broken stud.
Insert the screw extractor into the hole you drilled and turn it counterclockwise by hand to lock it into the hole.
Turn the extractor counterclockwise and remove the stud. Use care to ensure the axis of rotation is exactly perpendicular to the broken stud or you will most likely break the extractor.
Remove the head from the engine first to make drilling out the broken stud mistake free and easier at the same time. If you damage the threads while drilling, you will have to over-drill the hole after stud removal and install a thread repairing product such as Heli-coil.