Is there a way to unscrew a light bulb that has broken?
With the power off at the breaker, and verified with a non-contact tester, I've always just used a pair of needle nose pliers to grip the rim of the bulb base and turn it to remove. If the bulb is really stuck you might try spraying some WD40 around it.

A similar alternative, as others have mentioned in the comments, is to expand a pair of pliers inside of the base in order to grip the inside and turn it.