Bring a large pot of water to a boil and season generously with salt. This is your first chance to make the pasta taste great.
Add pasta and cook according to package directions. Four minutes before the pasta is done , add the broccoli florets to the boiling water. Drain, reserving 1/2 cup of pasta water.
While the pasta cooks, heat olive oil in a large skillet over medium-high. Pat chicken dry, season with salt and pepper, and cook in a single layer for 5–7 minutes until golden and cooked through. Transfer to a plate.
Reduce heat to medium. Add butter to the same skillet. Stir in garlic and red pepper flakes; cook 30–45 seconds until fragrant. Don't burn it—bitter garlic = sadness.
Pour in chicken broth, scraping up any browned bits. Simmer 2 minutes to reduce slightly. Add cream and bring to a gentle simmer until it thickens enough to coat a spoon, about 3–4 minutes.
Stir in Parmesan until smooth. Add lemon zest and 2 tsp lemon juice. Taste and adjust with more lemon, salt, and pepper. The sauce should be bright, savory, and lightly creamy.
Add cooked pasta, broccoli, and chicken to the skillet. Toss over low heat, adding reserved pasta water a splash at a time until glossy and well-coated.
Top with extra Parmesan, parsley, and a grind of black pepper. Optional: a drizzle of good olive oil for shine.