Mine usually break off inside and do not leave anything hanging out to weld to.
That's where the "Learn to use your welder" part comes in.
It took time for me to get the hang of it, but in the past 5 or 6 years that I've been using that trick I've removed hundreds of broken fasteners, and can count on one hand the number that didn't work and had to be drilled out.
A TIG welder is nice, that's what I have, but it can be done with a MIG, and even with a stick welder. Not as easy with the Stick, but if you must there are special welding rods for welding to broken bolts below the surface (I haven't used any, but a professional welder friend told me about them..)
Don't be afraid to try it, if you never try something you'll never get the hang of it.
Grigg