Remove broken cartridge of a Moen shower faucet
You should still be able to get the cartridge out.  In your bottom left picture, there's a little copper-colored tab sticking out (between 12 and 1 O'Clock, to the left of the threaded hole.)  You pull that out straight up and then the cartridge pulls out toward you.  I would think a pair of pliers would work for that.  Then what I do is take it to the hardware store and compare side-to-side and find the match.