Heat butter and oil in a large skillet over medium-high heat. Add peppers and onion, season with salt and pepper. Cook, stirring often, until almost tender, 10 to 12 minutes. Add zucchini and cook until peppers are tender and zucchini is crisp-tender, 4 to 6 minutes.
Meanwhile, cook pasta according to package directions. Reserve 1 cup pasta water; drain.
Add pasta, mozzarella, basil, and 1/2 cup of reserved pasta water to pepper mixture and toss to coat. (Add more pasta water if needed to create a sauce.) Season with salt and pepper.
Serve topped with mozzarella and basil.