Roll the dough out into finger-thick snakes. Using a knife, cut a round the size of a thumbnail and spread it out across the work surface with the knife (the cutting and spreading should be one motion), then flip the pasta with a flick of the thumb; doing so will make the center of the bit of pasta form up into a dome, and the orecchietta is done.
Continue making orecchiette until you have used up all of the dough.
Bring a large pot of water to a boil. Meanwhile, trim the broccoli (peel the stalks and crunch on them while you cook, if you like) and separate the crowns into large florets. Cut the florets into smaller ones, about 1/2-inch across. Set aside. Peel and finely chop the garlic and set aside.
Add the oil to a pan and then the broccoli, and a pinch of salt. Cook, stirring occasionally, until the broccoli turns bright green, about 3 minutes. Add the garlic and the red pepper flakes.
Add about 1/2 cup of the reserved pasta-cooking water to the broccoli. Cover, reduce heat to medium-low, and cook until the broccoli are tender. Add the drained pasta to the broccoli, toss to combine well. Add the anchovies and toss to combine. Add the cheese and toss to combine. Taste and add salt if you like. Serve hot, topped with more cheese.