Pasta Primavera

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1 pound dried rotini (corkscrew) or penne pasta

5 tablespoons butter

2 cups chopped red onions

1 cup chopped red bell pepper

2 teaspoons salt, plus more for the pasta water

1 teaspoon freshly ground black pepper

1 1/2 tablespoons minced garlic (about 5 cloves)

1 pound zucchini or yellow squash (or a mixture of both), halved lengthwise and cut crosswise into 1/2-inch half-moon pieces (about 4 cups)

2 cups broccoli florets, blanched

2 cups frozen mixed peas and carrots

1 cup diced canned tomatoes, with juices

1/2 cup finely grated Parmigiano-Reggiano cheese, plus more for serving

1/2 cup thinly sliced fresh basil, optional

1/3 cup extra-virgin olive oil