I've never tried this but I see no reason why it wouldn’t work., a conventional right-handed drill could screw the broken stud or screw further in. If you ground the shank off a high speed steel drill and re sharpened the now new cutting end and put the drill in reverse to drill the stud out, you may find the stud would start to unscrew on its own. I know you can buy left-handed HSS drills but its probably more convenient to make one.