Part identification: Board mount rotary switch
The broken part resembles the one on the right in this picture.

It  appears similar to this rotary encoder switch with a push switch. It's a 5 pin SMD type with three pins for the encoder and two for the push switch.

Quadrature pulses generated by the rotary encoder switch would be used for scrolling through the menu and the push switch for selecting an option, during set-up.