Bring a large pot of salted water to a boil. Add the rotini and cook until just tender, about 11 minutes. Drain, reserving 3/4 cup of the cooking water, and set aside.
While the pasta is cooking, melt the butter in a 14-inch saute pan over medium-high heat. Add the onions and bell pepper and cook until soft, about 4 minutes. Add the 2 teaspoons salt, black pepper, and garlic, and cook for 1 minute longer. Add the squash and continue to cook, stirring as needed, for 2 minutes. Add the blanched broccoli, frozen peas and carrots and cook for 2 minutes. Add the tomatoes and continue to cook, stirring, for 2 minutes.
Add the pasta and the reserved cooking water to the sauce and cook until the pasta is heated through and the ingredients are well combined, about 1 to 2 minutes. Remove from the heat and fold in the grated Parmigiano-Reggiano cheese and fresh basil, if using. Then drizzle with the extra-virgin olive oil. Garnish the dish with more cheese, if desired, and serve hot.