buy a small center punch, an easy out, some wd-40 or Liquid Wrench, a small drill bit about 1/8 diameter. Punch the center of the broken stud now drill on that spot about 1/4 inch deep with the 1/8 drill. Enlarge that drilled hole slightly with a 5/32nds or 3/16" drill to the same depth, maybe a little bit deeper. Now tap the easy out into that drilled hole, use a T-handle tap holder or a small crescent wrench with plenty of Liquid Wrench or WD-40 and slowly unscrew the broken stud. Make sure the EazyOut you have will work with a 3/16 size drill bit. Let the penetrating oil sit on the broken stud before doing all this for at least 2 hours and drench it. That will give it time to loosen up the grime or rust, etc that is holding it solid in there.
After you get the stud out just use a bolt instead of a stud. Use a stainless steel bolt if you can.