Three ways cap screws break:
1. Bottomed in the hole. Really bad, lots of force holding it from turning.
2. Rusted in. Heat, solvents and nut trick below are apt to work.
3. Over-torqued but not bottomed. The easiest, should be loose. Use an ice pick and hammer to apply torque to back it out.
The slickest broken bolt removal I've seen:
If the broken end is flush with the surface a welder can position a hex nut about centered on it and with a wire feed weld the nut on. Not only provides something to get hold of but the heat helps loosen. For that one guy who will want to rain on my parade - yes the welded bolt is junk and it's too short anyway so you throw it away and get a new one.