Preheat oven to 325°F. Heat oil in a large ovenproof skillet over medium heat. Add onion, frying pepper, mushrooms, garlic, 1/4 tsp. salt, oregano and 1/4 tsp. ground pepper. Cook, stirring often, until vegetables are tender and golden, about 10 minutes. Keep warm in oven.

While vegetables are cooking, split rolls and hollow out slightly. Place on a baking sheet with hollowed sides up and arrange 1 slice of cheese on each roll half. Crumble beef into a bowl and mix in remaining 1/4 tsp. salt and 1/4 tsp. pepper. Divide meat in half and then form into 2 oval patties, shaping meat to match shape and size of rolls.

Place rolls in oven to warm. Place a large heavy skillet over medium heat. Add beef patties to skillet and cook for 3 minutes; flip, place remaining cheese on top, cover pan, and cook until medium done, about 3 minutes.

Place roll bottoms on plates. Top each with some tomato sauce, a patty and some vegetable mixture. Cover with roll tops and serve.