To make the caramelized onions, heat the oil in a large nonstick skillet over medium heat. Add the onions – don’t worry if they’re piled high; they will cook down. Turn with a spatula so all the onions are evenly coated in oil. Continue to cook, stirring the onions every 8 minutes, until the slices have turned a dark, rich brown color, about 25 minutes. If the onions begin to burn, reduce the heat to medium-low and add more oil. Remove the onions and place them in a bowl.
Heat a large skillet over medium-high heat and add enough just oil to cover the entire bottom. After 2 minutes, reduce the heat to medium and place the patties into the skillet. Season with salt and pepper, and cook for 3 minutes. Flip, and cook on the other side for 1 minute.
Place 1 slice cheese on each patty and continue cooking for 2 minutes. Cover with a lid for the last 30 seconds to melt the cheese.
Toast the buns and set aside.
To assemble the burgers, place 1 patty on a toasted bun bottom. Top the patty with a tablespoon of caramelized onions, 1 lettuce leaf, 1 tomato slice, and a heaping spoonful of the South Lawn Herb Garden Mayo. Cover with the bun top.
Repeat with the remaining ingredients.
Wrap the sandwiches in wax paper. Let rest for 2-3 minutes and serve.