Bring a large pot of salted water to a boil. Cook pasta according to package directions until al dente, about 10 minutes. Drain and set aside, reserving 1 cup pasta water.

Meanwhile, cut broccoli into small florets. Heat 3 tablespoons butter in a large skillet over medium-high heat. Add broccoli in a single layer and cook undisturbed for 3 minutes until the bottoms turn golden brown. Stir and cook 2 more minutes until tender-crisp. Transfer to a plate.

In the same skillet, melt remaining 3 tablespoons butter over medium heat. Add garlic and cook, stirring constantly, for 1 minute until fragrant but not browned. Add red pepper flakes and stir for 10 seconds.

Add cooked pasta and broccoli to the skillet. Toss gently, adding pasta water a splash at a time (start with 1/2 cup) until the mixture is glossy and loosely coated. Season with salt to taste.

Remove from heat. Stir in parmesan cheese until melted. Squeeze half the lemon over the top, toss once more, and serve hot with extra parmesan and lemon wedges on the side.