Place rimmed baking sheet in oven. Preheat oven to 450 F, leaving pan in oven as it heats.
Cut broccoli into florets. Peel and thinly slice broccoli stems. In bowl, toss together broccoli florets and stems with oil, lemon zest, and salt. On hot baking sheet, arrange broccoli in single layer and return to oven; bake until browned and crisp-tender, about 15 minutes. Remove from baking sheet and set aside.
Reduce oven temperature to 375 F. On baking sheet, place walnuts and roast until a couple of shades darker, stirring once, about 10 minutes. Set aside.
Meanwhile, fill large saucepan with 8 cups water; bring to a boil. Add pasta; cook according to package directions until al dente. Drain in colander, rinse with cold water, and drain well.
In heavy-bottomed and preferably light-colored saucepan, heat butter over low heat until melted. Raise heat to medium and cook butter, stirring and scraping pan bottom often with rubber spatula, until butter foams, the color turns golden-brown, and it smells nutty. Do not let butter turn dark brown, which means it has burnt. Take pan off heat, stir in chili flakes, and immediately transfer butter and any browned milk solids to heatproof bowl to cool.
Divide pasta and broccoli among serving plates, sprinkle on walnuts, and drizzle on browned butter. Squeeze on lemon juice and then season with pepper and cheese.
Recipe Notes
Per serving: 449 calories; 20 g protein; 27 g total fat (8 g sat. fat); 41 g total carbohydrates (4 g sugars, 9 g fiber); 481 mg sodium