Trim bottom 1" from stem of 1 large head of broccoli (1–1¼ lb.); discard. Peel tough outer layer from stem as needed; discard. Finely chop stem; cut broccoli head into 1" florets.

Using a Microplane, finely grate zest from ½ lemon into a small bowl and cover with a damp paper towel. Slice lemon into thick rounds and remove seeds; discard. Very finely chop lemon rounds (peel, pith, and flesh); set aside.

Cook broccoli stem in a large pot of boiling salted water, about 3 minutes. Add broccoli florets and cook until deep green and very tender, about 5 minutes. Using a spider or slotted spoon, transfer broccoli to a colander; reserve pot of water.

Return water to a boil. Add 12 oz. short pasta (such as mezze rigatoni or cavatappi) and cook until very al dente, about 2 minutes less than package directions. Drain, reserving 2 cups pasta cooking liquid.

Cook 8 oil-packed anchovy fillets, 4 garlic cloves, thinly sliced, and ⅓ cup extra-virgin olive oil in a medium Dutch oven or other heavy pot over medium, swirling pan occasionally, until garlic is light golden and anchovies have dissolved, about 3 minutes. Stir in ½ tsp. crushed red pepper flakes and reserved chopped lemon, season with freshly ground black pepper, and cook 1 minute.

Remove from heat and add broccoli to pot. Using a wooden spoon or a firm heatproof rubber spatula, smash broccoli into very small pieces. Add pasta, 1 cup pasta cooking liquid, and ¾ cup finely grated Parmesan and toss aggressively, adding more pasta cooking liquid as needed to loosen, until pasta is coated in sauce. Season with kosher salt.

Divide pasta among shallow bowls. Drizzle with more oil and top with reserved lemon zest and more red pepper flakes and remaining ¼ cup finely grated Parmesan.