My preferred technique for removing a rusted and snapped-off fastener is extreme heat. I accomplish this by welding a new nut to the top of the stud. The heat causes the metal to expand which breaks the corrosion bond between the stud and the surrounding metal. (Why doesn't heating the stud cause it to expand and bind in the surrounding fixture? I don't know. Maybe the expansion isn't so much as to cause binding.) The newly attached nut gives a shape that's easy to grab with a wrench -- much less chance of slipping as compared to locking pliers on the broken stud.
Choose a nut that's big enough to slide easily over the stud. There's no scale in the photo but I'm going to guess 5/16" or 3/8" would do for this job. Position it so that the stud rises no more than half way through the nut. Build a molten pool on top of the stud, then pull the weld pool over into the nut and fill until it's plugged. I use a MIG for this because it's what I have. I think TIG or oxy-fuel would work but stick welding would be challenging at this scale.
Helpful tips: Bring several nuts. It's easy to get the nut filled with weld but have poor penetration/fusion with the stud, so it may take a few tries. Pre-soaking with a penetrating oil could help, but if you do, you'll have to clean the surface with water and detergent, alcohol, etc so that the oil residue doesn't spoil the weld.