That sucks. There are few things tougher than getting a broken off stud with an ez-out broken off in it. Sometimes, you can have some luck if you drill a very small hole just off to the side of the ex-out. It can release the stud's hold on it and let you dig the ez-out out of there without too much damage to the surrounding area. Good luck! I had to basically grind away the stud and surrounding area on a part once, have it welded to build it back up and then machined to get it flush again.

There was this guy in town many years ago, who had a business of removing things like this. He used a machine that uses electicity to zap little bits of the metal away until there is nothing left of the ez-out. It was like an electric drill bit without the bit.

Once this is past you, buy a quality set of left-hand cobalt drill bits.