Bring a large pot of salted water to a boil. Cut the broccoli crowns into large chunks. Drop the broccoli into the boiling water and cook for 5 minutes, until tender. A paring knife should slide right into the stem with no resistance. Scoop the broccoli out with tongs or a strainer; keep the water boiling. Rinse the broccoli in a colander with cold water. Drain on paper towels, gently squeezing out excess water. Chop the broccoli into small pieces.
Add the pasta to the pot.
Meanwhile, heat the oil in a large skillet over medium heat. Add the garlic, anchovies, and red pepper flakes. Cook, stirring and mashing the anchovies with the back of a wooden spoon, for 3 to 4 minutes until the garlic is golden. Raise heat to medium-high. Add the broccoli and a ladleful (about 1 cup) of pasta water to the pan, and cook, stirring for about 5 minutes, adding more water as needed to achieve a cohesive sauce. Add the olives and add salt and pepper to taste.
Scoop some more pasta water from the pot. Drain the pasta and add to the pan and stir to coat, again adding more pasta water as needed to help the sauce coat the pastaevenly. Add the butter and stir until melted.
Divide among four bowls and serve with grated cheese and red pepper flakes on the side. Enjoy!