1 teaspoon minced fresh parsley, for garnish (optional)
Melt the butter and oil in a large nonstick skillet over medium heat. Once the butter is melted, add the mushrooms, salt, and black pepper, and cook until the mushrooms are browned, about 12 minutes, stirring occasionally.
While the mushrooms cook, cook the burgers. Shape the meat into 4 patties. Preheat a large cast iron skillet over medium-high to high heat. Once hot, add the patties and turn the heat down to medium to medium-high. Cook the burgers for 4 minutes on each side, flipping once.
Once the burgers are cooked, place 1 slice of cheese on each burger. Turn off the heat, drape a piece of foil on top of the burgers, and let the cheese melt.
Serve each burger on a bed of 2 lettuce leaves and top with mushrooms. Garnish with parsley, if desired.