Broken Piece of Rusted Pipe

I have ran into this countless times on the farm and during service work and the simplest way I have found is to either cut the broken section with a hack saw blade end or a file and then just fold the broken piece inward with a point punch then just twist it out with a pliers.

You don't always have to cut all the way through the bad piece. Just giving it a weak spot for it to fold at is enough. If you cut to deep you will cut up too much thread surface of the part your trying to get it out of and then you will be far more likely to have a leak.

Sometimes you can get lucky with using an extractor tool but typically if the part rusted off or was stuck in so bad it twisted off cutting and folding it in is the most effective and least frustrating way to get it out.

You could weld something to it. I haven't tried it on ceramic but on engines it works a treat, eg: a broken stud. Put a nut over the stub, weld it up, the heat loosens the stud and it comes out with a spanner on the nut.

There are numerous ways; it would be helpful if you post a picture to show the exact problem.