Heat the olive oil and butter in a large, heavy skillet over medium-low heat. Add the onions to the skillet. Cook until golden brown and very tender, about 15 to 20 minutes, stirring frequently. Remove the onions from the pan.

In a bowl, mix the ground beef with the burger seasoning. Shape the ground beef into four 4-ounce patties. Arrange the burgers in the same skillet and cook over medium heat until cooked through, or about 5 to 6 minutes on each side.

Meanwhile, toast the buns on a griddle or grill pan; set aside.

Top each burger with 2 slices of cheese and cover the pan until cheese has melted. Place a lettuce leaf on the bottom of a toasted bun. Top with a cheeseburger, then top with a generous amount of caramelized onion and the top of the bun.

Serve with condiments and sliced tomatoes on the side.