1. Fill a large pot with generously salted water and bring to a boil. Add the pasta and cook according to package directions. After 4 minutes, toss the broccoli into the pot. Reserve 1 cup of the cooking water, then drain the pasta and broccoli in a colander.

2. Meanwhile, heat the oil in a large skillet over very low heat. Add the garlic and sauté just until it begins to sizzle (do not let it brown). Turn the garlic and continue to cook, 1 to 2 minutes. Add the salt, pepper, paprika, pepper flakes, and salt and black pepper to taste, and stir to arrest the browning. If the garlic appears to be coloring, stir it sooner. Continue to cook until the smaller pieces of garlic become golden. Turn off the heat and set the skillet aside.

3. Once the pasta and broccoli are cooked, turn the heat on under the skillet to medium-low. Add the pasta and toss to thoroughly combine with the oil and garlic. Season to taste with the salt and pepper. Moisten the pasta, if necessary, with the reserved pasta water. Garnish with cheese and serve immediately.