Using your hands, remove broccoli crowns one by one. Rinse and set aside. If the leaves are still green and undamaged, wash them too.

In a medium saucepan over medium heat, add two tablespoons of olive oil and saute the garlic, anchovies, and chili flakes, until the anchovies have dissolved and the garlic is golden and fragrant. Add the tomato paste, stirring until bright red, around 1 minute.

Add the tomato puree and 5 cups of the water, along with the kosher salt and black pepper to taste. Once it comes to a boil, add in the broccoli.

Let the broccoli cook over low heat for about 35 minutes, stirring every 10-15 minutes. Using a fork, test to see if the broccoli is done.

When the broccoli has softened, add in the pasta and the remaining one cup of water. Stir together and cook for 10-15 minutes, on a low flame, stirring often, until the soup has thickened and pasta is fully cooked. Season to taste with more salt as needed. 

To serve: Drizzle the pasta mixture with olive oil, and sprinkle with fresh parsley and chili flakes. Shave fresh pecorino on top.