Green’s Dictionary of Slang

to look at fixedly, to stare at; to ogle.

1937		E. Anderson Thieves Like Us (1999) 9: He had opened up an account in a bank just to big-eye it good.
1950		Goldin et al. DAUL 27/1: Big-eye, v. To look at with covetousness, wonderment, fear, etc.
1958		E. Gilbert Vice Trap 122: A car went across [...] The people in it big-eyed us.
1968		D. Ireland Chantic Bird 148: She big-eyed him and went on slobbering.
2003		Dolphin Yacht Club 🌐 [pic. caption] Mike Keating big eyeing a burger!