In my experience, an "Easy-Out" like you pictured will indeed do the job; just not the way you think it will.......
See, once you get the screw extractor screwed into the shank of the broken bolt, and you're cranking on it as hard as humanly possible; it will
snap off, flush with the offending bolt-shank. Don't worry, they're
really designed to do this. Then, when in desperate frustration, you drag the cutting torch out to melt the remains of the "easy-out"; you'll find that it's actually harder and less prone to melting than the metal composition used in the bolt's manufacture. You'll end up with a white-hot-but-still-solid chunk of "easy-out" falling out of the hole; surrounded by molten globs of what used to be the bolt. Then you can more easily drill and tap the hole; as 96BlueStangGT suggests.
At least, that's been my experience.......