Bring a large pot of water to a rolling boil. Add 1 tablespoon salt and the spaghetti. Cook according to package directions until al dente (about 9 minutes). Reserve 1 cup of pasta water, then drain and set aside.

While the pasta cooks, bring a separate medium pot of water to a boil. Add broccoli florets and cook for 2 minutes until bright green and slightly tender. Drain and immediately rinse with cold water to stop the cooking. Set aside.

In a large skillet over medium heat, heat 1 tablespoon olive oil. Add minced garlic and red pepper flakes. Sauté for 30 seconds until fragrant, then add the blanched broccoli. Cook for 2-3 minutes, stirring occasionally, until the broccoli is tender and slightly charred at the edges. Transfer to a plate.

In the same skillet, add the remaining 1 tablespoon olive oil. Pat the shrimp dry and season with salt and pepper. Add shrimp in a single layer and cook for 2 minutes per side, until pink, opaque, and reach an internal temperature of 145°F (63°C). Return the broccoli mixture to the skillet along with the cooked pasta.

Add lemon zest, lemon juice, Parmesan, and 1/4 cup reserved pasta water to the skillet. Toss everything together over low heat until the sauce coats the pasta evenly. Add more pasta water a tablespoon at a time if it seems dry. Adjust salt, pepper, and red pepper flakes to taste.

Divide among four bowls, top with chopped parsley, extra Parmesan, and a lemon wedge. Drizzle with a tiny bit of olive oil if desired. Serve immediately.