Pour the olive oil into a large skillet over a medium-high heat. When the oil is hot, add the bell pepper and onion, then sprinkle with a pinch each of salt and pepper. Cook, stirring frequently, until lightly browned and just tender, 8 to 10 minutes. Add the garlic and cook until just softened, about 1 more minute. Stir in the pizza sauce and 1/2 cup water, then simmer until thickened but still saucy, about 5 minutes. Set aside and keep warm.
Combine the beef, sausage, oregano, sage and remaining teaspoon each of salt and pepper in a large bowl. Mix with your hands until well combined. Form into six patties, about 1 inch thick and 4 1/2 inches wide.
Grill the patties until cooked through, 6 to 7 minutes per side. Top each with a slice of cheese during the last 1 to 2 minutes and cover to melt.
Serve the burgers on the rolls and top with the pepper and onion sauce, then add the fresh basil.