Boil the pasta. Bring a large pot of salted water to a boil. Cook the pasta until just shy of al dente, according to package directions.

Blanch the broccoli. During the last 2–3 minutes of pasta cooking, add the broccoli florets to the pot. Cook until bright green and crisp-tender, then reserve 1 cup of pasta water and drain both pasta and broccoli.

Start the sauce. In a large skillet, warm the olive oil and melt the butter over medium heat. Add the garlic and cook 30–60 seconds until fragrant, not browned.

Make it creamy. Pour in the heavy cream and a pinch of salt and pepper. Simmer gently for 2–3 minutes, stirring, until slightly thickened.

Add the cheese. Reduce the heat to low. Stir in the Parmesan a handful at a time until smooth and glossy. If using mozzarella, add it now. If the sauce is too thick, splash in a bit of the reserved pasta water.

Toss it together. Add the drained pasta and broccoli to the sauce. Toss until everything is coated. Adjust with more pasta water for a silky finish.

Brighten and season. Add lemon zest and a squeeze of juice. Taste and season with more salt, pepper, and red pepper flakes if you like a hint of heat.

Serve. Plate immediately with extra Parmesan on top. The sauce thickens as it cools, so enjoy it warm.