Combine the ground beef, sausage, garlic, oregano, 1/4 teaspoon salt and a few grinds of pepper in a medium bowl; mix with your hands until just combined. Form into four 3/4-inch-thick patties; set aside.
Preheat a grill pan or grill over medium heat. Toss the onion and bell peppers with the olive oil, 1/4 teaspoon salt and a few grinds of pepper in a bowl. Grill the vegetables, turning occasionally, until lightly browned and tender, about 7 minutes. Transfer to a plate and set aside.
Brush the grill pan with olive oil. Add the burgers and cook until browned and cooked through, about 4 minutes per side, topping with the cheese and covering during the last 2 minutes.
Serve the burgers on the buns and top with the onion-pepper mixture, basil and marinara sauce. Serve with chips.