Identify me! Shower temperature regulator?

No idea what this piece is or is called, but I assume it regulates the mix of hot and cold water in my shower. We bought the house two years ago. A week ago, shower went to only medium cold water; all other faucets were fine. I took apart what I could, found this little fella in there with the crack in it, put it back in and then only got hot hot water no matter what.

Took it apart today again to take a picture of this guy and now we are only getting medium hot water.

I am unsure of the brand of faucet we have, but it’s a single “pull out” style knob that turns to either direction. It does have a logo that looks like a diamond but the sides kinda curve inwards. I’m assuming that I need to replace this part specifically - does anyone know what this part is called and a specification for it (if that’s even possible with the info I have)? Thanks!