Whether it is a broken tap or screw, the best way to remove it is in a mill using a bottom-cutting carbide milling cutter. This way you can find its center without risk of the bit wandering off and ruining the workpiece. As Phil pointed out, you'll want to first knock off any projections; even then you risk breaking the cutter. If you don't have a mill, take it to a competent machinist who will not scratch anything up.
