Cook the pasta in a large pot of salted boiling water, stirring occasionally. After 5 minutes, add the broccoli and cook 4 minutes more. Drain, saving 1 cup of the pasta water.
Meanwhile, heat the olive oil and butter in a large skillet over medium heat and cook the garlic, anchovies and red pepper flakes 5 minutes.
Add the broccoli, pasta, salt and pepper and toss, adding pasta water as needed for a light sauce. Sprinkle with Parmesan to serve.