Preheat the broiler. Place peppers on a broiling pan. Broil, turning frequently, until skins blister and turn black. Transfer the peppers to a bowl, cover with plastic wrap and set aside for 10 minutes.
To peel peppers, place under cool running water and peel away blistered skins. Slice open and wash out seeds. Place the peppers in a food processor or blender; add balsamic vinegar, crushed red pepper flakes, salt and pepper. Puree until smooth.
Bring a large pot of salted water to a boil. Add broccoli florets and cook for 1 minute. Scoop out the broccoli with a slotted spoon and set aside. Bring the water back to a boil.
Cook pasta until al dente, about 8 minutes.
Meanwhile, spray a large skillet with cooking spray and place over medium-low heat. Add garlic. Cook, stirring, for 30 seconds. Stir in tomatoes and increase the heat to medium. When the tomatoes begin to simmer, stir in pepper puree and broccoli; cook for 2 minutes. Stir in basil and remove from the heat.
Drain the pasta and add to the skillet, stirring to coat.
Spoon into shallow bowls and garnish each with 1 Tbsp grated Parmesan.