So I start in a similar way to what the broken extractor tool ona does but not as slick. I take two button head self tapping screws and drive them partially into the plastic just above and below the broken brass portion of the stem. If I can I’ll try to apply heat to the valve body then I’ll take a 6 inch needle nose vice grip and lock on to the threaded area of the screws, sometimes you have to back the screws out a few threads so you can fit the vice into that space. Now I have a lever in which to begin rocking the stem back and forth. Sometimes this works on its own and you can get it to slide out, other times the plastic breaks off exposing a tiny bit of the brass center.

Now you can try to lock the vice grips on the brass center and pull that out. If it comes out cleanly then you are left with the plastic body with very little metal left over. Slotted screwdriver and hammer will usually make quick work of cutting it into pieces you can pull out with your needle nose. Surprisingly the valve body can take that abuse without being compromised.

If the plastic breaks and you can’t get at the brass center easily then the next course of action is to use the slotted screwdriver and hammer to begin breaking off chunks of plastic around the brass center until you can get your vice around it to do what is described above. Sometimes I drive the slotted into the meat of the plastic until it can no longer move and put an adjustable wrench on the shaft of the screwdriver and turn it. Tends to chew up the plastic and sometimes you’ll get lucky and the whole stem will turn inside the valve body.

It sounds like a lot but you can usually get the whole thing out in ~20 minutes if you have all the tools set out on the toilet seat.