@G-P "Stuff your face" is a fair equivalent, and even has the same connotations of eating for the sake of eating, or gluttony.
Since I grew up in Hicksville, USA 🇺🇸, I've also heard "fill your craw," and "put <this, that> in your gullet," which are both bird related. Though, to be honest, even in my own head they sound like something from a trashy western.