Bring a large pot of well-salted water to a boil. Add pasta and cook according to package directions until al dente. In the last 2 minutes of cooking, add the broccoli florets to the pot. Drain and set pasta and broccoli aside.

In the same pot over medium heat, melt butter. Whisk in flour and cook 1–2 minutes, whisking constantly, until the roux smells slightly nutty.

Slowly whisk in the milk until smooth. Simmer 3–5 minutes, whisking occasionally, until the sauce thickens enough to coat the back of a spoon.

Whisk in garlic powder, onion powder, salt, and black pepper. Reduce heat to low.

Add cheddar (and mozzarella, if using) a handful at a time, whisking until fully melted and smooth. Stir in Parmesan and cream cheese or a splash of cream if using; adjust thickness with a spoon of milk if needed.

Fold the cooked pasta, broccoli, and chicken into the cheese sauce until evenly coated. Taste and adjust seasoning with additional salt and pepper.

Optional finish: transfer to a broiler-safe dish, top with extra cheese and/or breadcrumbs, and broil 2–3 minutes until golden and bubbly. Serve warm.