mystery part

Hi Guru's! We're stumped and would appreciate some help. We have an older Delta setup in our shower with a handle that broke off recently. The handle/cartridge assembly [apologies if this is the wrong vernacular!] is what we are looking to replace, but we cannot find it anywhere...

This came out of a shower and I need to know what it's called, no one seems to know. Would like to be able to replace it. Tia