Bring a large pot of water to boil over high heat. Add salt and then add the pasta. Cook according to the package directions. After 6 minutes, add the broccoli to the pot with the pasta. Once the pasta is al dente, reserve 1 cup of the cooking water, then drain the pasta and broccoli in a colander.
Meanwhile, heat the oil in a large skillet over very low heat. Add the garlic and saute just until it begins to sizzle (do not let it brown). Turn the garlic and continue to cook, 1 to 2 minutes. Add the paprika and season with salt and pepper; 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.
Add the pasta and broccoli to the garlic oil. Toss together until combined. Add the butter and stir to melt. Add the cheese and stir again. Moisten the pasta, if necessary, with the reserved pasta water. Season to taste with salt and pepper. Garnish with cheese and serve immediately.