Seeking advice on how to extract broken lug bolts

Someone really GOOD with a TIG could go in there and build those broken bolts up flush and weld nuts on and back them out. I have a friend who is that good in Macon. I've seen him get some out of heavy equipment at a rock quarry that were broken off over an inch deep. I was amazed.
I don't know why this idea wouldn't work with a MIG welder too. The heat cycles will help too.