LAX isn't a gourmand's destination. But not far from LAX, someone may find many great breakfast sit-down restaurants. However, most of those don't open until 8am so, depending upon your priorities, you may want to breakfast before LAX.

And Proposition 12 will ensure the eggs were cage-free and here are the regs about eggs: https://www.cdfa.ca.gov/AHFSS/mpes/esqm.html