Heat broiler. Arrange fennel and chiles on one rimmed baking sheet and tomatoes and onions on another. Toss each with 2 tablespoons oil and season with salt. Broil 3 minutes, then toss and continue broiling until vegetables are tender and charred in spots, about 2 minutes more. Transfer vegetables and pan juices to a bowl.
Cook pasta in salted boiling water according to package instructions. Reserve 1 cup cooking water, drain, and return to pot.
Add vegetables, Parmesan, and reserved water. Stir in basil and remaining oil and adjust seasoning. Sprinkle with additional Parmesan.