Put sliced onions in the pan and cook on low for 20-30 minutes. Stir occasionally. Onions will begin to turn brown.
Pour 1/3 cup water over onions and stir. Simmer on low till water cooks off entirely. Turn off heat and let sit till ready to put on burgers.
The onions may also be sliced, coated in avocado oil and grilled. Once grill marks are in place move to a grill safe plate or pan. Continue to cook while burgers cook.
Burgers
Mix together hamburger, onion soup mix reserving one teaspoon, Worcestershire sauce, parmesan cheese, and egg with hands.
Form hamburger combination into four patties and place on grill over medium heat. Flip burgers only once to keep them from falling apart. You will want to cook for about 8 minutes before flipping.
Mix the remaining soup mix with a half cup of mayonnaise. Spread the french onion aioli on buns or toasted Italian bread.
Top burgers with cheese for the last minute before removing from grill.
Place burgers on toasted bread, top with caramelized onions and another piece of bread.