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I've had to remove both large and small taps after they broke off at surface level. For the large one ( 1-1/4" bolt tap ) I used a very dull air chisel bit against a small piece of aluminum bar. And stopped several times to use compressed air to blow out anythingthat would come out.
I held the aluminum and chisel joint in one hand (with glove) it gets very hot. the other end of the aluminum rotated the tap.With small ones, I turn the work piece upside down if possible and hit it with lots of air before trying to turn the tap back out. Once you get some of the small chips out of the way the tap has room to move back out. Again, and aluminum punch (home made of course) works best because it catches the sharp edges of the broken tap.