1. Preheat oven to 450°F; line a large, rimmed baking sheet with foil. Combine broccoli, 1 Tbsp. olive oil and Italian seasoning in a large bowl; toss to coat. Place broccoli in a single layer on baking sheet. Roast, stirring once or twice, until broccoli is crisp-tender and beginning to lightly brown, about 15 minutes.
2. Bring a pot of salted water to a boil. Cook pasta until al dente, about 4 minutes or as package label directs. Drain, reserving 1 1/2 cups pasta cooking water.
3. Warm 3 Tbsp. olive oil in a Dutch oven over medium heat. Sauté garlic until just golden brown, about 2 minutes. Remove from heat; stir in crushed red pepper, 1 tsp. salt, 1 1/4 cups reserved pasta water and remaining olive oil. Add pasta and 1/2 cup cheese; simmer over medium heat, tossing gently, until sauce thoroughly coats pasta (add remaining 1/4 cup pasta water if needed), 2 to 3 minutes. Remove from heat and toss in broccoli, remaining cheese and basil. Serve immediately.