Don't even think about using stud extractors. They're worse than useless.
If there's sufficient stud poking out, you might be able to get a grip on it with mole grips, but it's probably siezed in there reasonably tightly by products of galvanic corrosion (mostly oxides of aluminium) otherwise it would have unscrewed rather than breaking in the first place. So unless you loosen it with heat and gentle impacts, the mole grips will probably only chew the end of it up without shifting it.
Rather than weld a bar onto it, place a nut over what's left of it and weld the inside of the nut to the broken stump. It's a bit of a fiddly weld, especially if the head is still on the bike, but do-able. I find the nut works better than bar because it contains the weld pool, you get a stronger connection to the stud, and you can get a spanner on it when it's done. If it's broken off close to or flush with the alloy head, you can bore a countersink into the nut to make the access for welding a bit easier. You could do it with a stick welder using a small welding rod, but best to use MIG, TIG or oxy welder to avoid filling up the inside of the nut with slag. It's worth the time investment to do a good weld if you end up with a successful outcome. As grant said, the heat from the weld will break down the corrosion oxides that have caused it to sieze, and subsequent shrinkage when it cools will more than likely loosen the stud and it will come out easily.
I've had 100% success with the above method, but if it fails, you may need to take the head to an engineering shop to have it spark eroded out.
Good luck
Cam