Heat oven to 400° F. Cook the pasta according to the package directions. Reserve 3/4 cup of the cooking water, drain the pasta, and return it to the pot.
Meanwhile, on a rimmed baking sheet, toss the broccoli, walnuts, oil, garlic, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Roast, tossing once, until the broccoli is tender, 18 to 20 minutes.
Toss the pasta with the broccoli mixture, butter, and 1/2 cup of the reserved pasta water. (Add more water if the pasta seems dry.) Sprinkle with the Parmesan before serving.
Tip: The combination of roasted broccoli, walnuts, and garlic also makes a delicious side dish for chicken, beef, or pork.