Bring a large pot of salted water to a boil and cook pasta according to package instructions. When pasta is al dente, reserve 1 cup of starchy cooking water and drain the rest.
While the pasta is boiling, heat the butter and olive oil in a large skillet over medium heat.
Once hot, add the onion and cook for 5 minutes, or until soft and translucent. Add the garlic and cook for 30 seconds more, until fragrant.
Add the pepper, zucchini, squash, and asparagus to the skillet and continue cooking for about 5 minutes to soften the vegetables.
Add the tomatoes, salt, and pepper and continue cooking for 2 minutes.
Add the pasta, 1/3 cup pasta water, and Parmesan to the skillet with the vegetables over low heat and toss to combine. Continue tossing until the cheese has melted and you have a glossy sauce that clings to the pasta. You may need to add additional pasta water, a tablespoon or two at a time to reach the desired consistency.
Tear the basil leaves into small pieces and combine with the pasta before serving.