Well if you don't want to get left-handed drill bits - something always recommended but something I've never gotten despite it being a good idea, you can probably grab the rest with vice grips and remove it. They aren't in there that tough.
I personally use a stud remover, 3/8" is a pretty good surface to grab onto.
Of course in the past when I completely wrecked a stud, I panicked and did quite a number on the hole - don't do this. You can look it up, it was posted on this very forum. Embarrassing to say the least.
