Heat 2 tablespoons olive oil in a large skillet over medium heat. Add the sausage and cook, breaking up chunks with a wooden spoon or potato masher, until browned, 7 to 9 minutes.
Meanwhile, heat the remaining 2 tablespoons oil in a second large skillet over medium heat. Add the onions and bell peppers. Season with salt and pepper. Cook, stirring occasionally, until the vegetables are tender and cook down, about 10 minutes. Add the garlic and cook until fragrant, about 30 seconds. Add the vegetables to the browned sausage. Pour the balsamic vinegar into the skillet and let cook over high heat until it becomes a syrupy glaze, 3 to 5 minutes. Remove from the heat and stir in 1/4 cup chopped basil. Add to the sausage.
Meanwhile, open the rolls and place them on the prepared sheet pan. Place a slice of cheese on the top and bottom of each bun. Broil until the cheese melts, about 2 minutes. Remove from the oven and top the cheesy bottom buns with the sausage and pepper mixture. Sprinkle with some of the remaining basil. Smush on the cheesy top buns and take a messy bite.