I have done that same thing in the past and that is the worse possible outcome.

My first duty station in the Navy was overhauling various antennas. I learned the hard way about broken bolts, extractors and corrosion.

They make an extractor remover, never tried one though. I usually used a punch to fracture the broken extractor, cause you ain't drilling it out. I understand that some machine shops can burn it out with a fancy machine, never used that either. But it'll cost at least a head gasket as well.

I never tried heating the broken extractor up red hot and hitting it with a freeze spray, but some have had success.