Set EGG for direct cooking (no convEGGtor) at 400-450°F/204-232°C.
Form beef into 4 burger patties. Sprinkle all over with salt and pepper.
Swirl oil in the 12-inch Lodge Cast-Iron skillet. Layer in onion and peppers and place skillet on the hot grill surface, slightly to one side. Cook until the vegetables are sizzling in the oil, about 3 minutes. Stir the vegetables and continue cooking, stirring often, until they are soft, about 12 minutes.
Meanwhile, when there is about 4 minutes left for the peppers, place burgers on the EGG next to the skillet. Cook, rotating ¼ degrees after 2 minutes for a total of 4 minutes on the first side.
Scrape peppers and onions to one side of the skillet with a spatula. Flip two burgers onto their uncooked side on the plain side of the skillet. Use tongs or the spatula to top the burgers with half of the pepper mixture, dividing evenly. Repeat clearing spots and topping with the remaining two burgers. Top with three slices of cheese per burger. Close the EGG and let cook until the burgers are cooked to desired doneness and the cheese is melted, 4 to 6 minutes. Serve on buns.
Makes 4 servings