Boil the pasta: Bring a large pot of salted water to a boil.
Cook pasta until just shy of al dente. Scoop out 1 cup of pasta water, then drain.
Blanch the broccoli: During the last 2 minutes of pasta cooking, add the broccoli florets to the pot. Drain with the pasta.
This saves time and one extra pan.
Season the shrimp: Pat dry. Toss with 1/2 teaspoon salt, a few grinds of pepper, and a pinch of red pepper flakes.
Sauté the shrimp: In a large skillet, heat 2 tablespoons olive oil over medium-high. Add shrimp in a single layer.
Cook 1–2 minutes per side, just until pink and opaque. Remove to a plate.
Build the garlic sauce: Lower heat to medium. Add another tablespoon of olive oil and the butter.
Stir in garlic and a pinch of red pepper flakes. Cook 30–60 seconds, until fragrant but not browned.
Deglaze and simmer: Add broth (or 1/2 cup pasta water) and scrape up any browned bits. Simmer 2 minutes to reduce slightly.
Combine: Add pasta and broccoli to the skillet.
Toss with the sauce, adding splashes of reserved pasta water as needed to make it silky.
Finish: Return shrimp to the pan. Add lemon zest, 1–2 tablespoons lemon juice, and parmesan. Toss until the cheese melts and everything is glossy.
Adjust salt, pepper, and heat with more red pepper flakes.
Garnish and serve: Sprinkle with parsley and an extra drizzle of olive oil. Serve hot with more parmesan on the side.