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Bring a large pot of salted water to boil and add the broccoli florets. Cook until tender, then remove with a slotted spoon. Add the pasta and cook until al dente, reserving ½ cup cooking water before draining.
In a large, heavy skillet, heat olive oil over medium heat until shimmering. Add garlic and red chile flakes and cook until fragrant, about 1 minute, then add broccoli and toss well in the oil, cooking until completely tender, 1-2 minutes.
Add pasta to the pan with the broccoli and toss well to coat. Add half the Parmesan and reserved pasta cooking water little by little until it creates a sauce that clings to the noodles. Season to taste with salt and pepper. Serve immediately with remaining Parmesan.