Bring a large pot of salted water to a boil. Add the fusilli and cook as the label directs. Add the sugar snap peas and/or broccoli, carrots and bell pepper to the boiling water during the last 2 minutes of cooking. Reserve 1/2 cup cooking water, then drain the pasta and vegetables and return to the pot.
Meanwhile, heat the olive oil in a large skillet over medium heat. Add the garlic and cook until just golden, about 30 seconds. Add the tomatoes, red pepper flakes and 1 teaspoon salt; cook until the tomatoes begin to wilt, about 2 minutes. Stir in 1/4 cup of the reserved cooking water. Pour the tomato mixture over the pasta and vegetables. Add the mint, parmesan and half the goat cheese and toss to combine. Season with salt.
Divide the pasta among bowls. Top with the remaining goat cheese and drizzle with olive oil.