Preheat a grill in 2 parts to low and medium heat. Cover and grill the bacon over low heat, turning halfway through, until crisp, 10 minutes; let drain.
Divide the beef into quarters. Enclose a piece of cheese in each quarter; pat into four 4-inch patties. Brush with EVOO and season with salt and pepper. Cover and grill the patties over medium heat, turning once, until browned, 5 minutes over medium. Grill the guns over low heat until toasted, 2 minutes.
Assemble the burgers on the buns with the mayo, tomato, lettuce and bacon.
