Meanwhile, heat 2 tablespoons oil, garlic, and pepper flakes in medium saucepan over medium heat until fragrant but not brown, about 2 minutes. Stir in crushed and diced tomatoes, bring to simmer, and cook until slightly thickened, 15-20 minutes. Stir in 3 tablespoons basil, sugar and 1/2 teaspoons salt.
Add 1 tablespoons salt and pasta to boiling water. Cook until al dente. Reserve 1/4 cup cooking water, drain pasta, and transfer back to pot. Stir in sauce and remaining tablespoons oil, and cook over medium heat until heated through (adding reserved water if pasta seems dry), about 1 minute. Stir in mozzarella and season with salt. Serve, sprinkling with remaining 2 tablespoons basil and Parmesan.