[1project2many]

Is this a steel plug? If you have an oxy / acetelyne torch I strongly recommend as the first approach heating the plug (not the head) to red then letting it cool. After the plug has cooled try removing it with an easy out. This trick works amaingly well, even with plugs that are not broken.

The type of extractor makes a difference. Tapered extractors that thread in when turned counter clockwise are the least likely to work. I prefer straight extractors that are driven into the broken fastener.

You can sometimes use a carbide burr to thin the plug, making a notch. Grind extremely close to the threads then use a punch to tap a portion of the plug inward, compressing the plug to remove tension.

If you have an air hammer and patience it is sometimes possible to use the air hammer with the "mushroom" tip to gently tap, tap, tap, tap on the plug loosening it in the head.
Good luck...