Bring a large pot of salted water to a boil. Cook the penne pasta according to package directions until al dente. During the last 3 minutes of cooking time, add the broccoli florets directly to the pasta water to blanch them until tender-crisp.

Before draining, reserve 1 cup of starchy pasta water in a measuring cup. Drain the pasta and broccoli together in a colander and set aside.

Heat the olive oil and 2 tablespoons of butter in a large skillet over medium-low heat. Cook the diced onion for 4-5 minutes until softened and translucent. Add the minced garlic and cook for just 30 seconds until fragrant, being careful not to let it brown.

Pour in the heavy cream and chicken broth. Bring the mixture to a gentle simmer for 2-3 minutes. Look for small bubbles around the edges rather than a rolling boil to prevent the cream from breaking.

Remove the skillet from the heat completely. This is a crucial step to prevent graininess. Whisk in the Parmesan and mozzarella cheeses until the sauce is completely smooth. Season with Italian seasoning, garlic powder, salt, pepper, and red pepper flakes.

Add the drained pasta and broccoli to the creamy sauce, tossing gently with tongs to coat evenly. Fold in the shredded rotisserie chicken. If the sauce is too thick, add the reserved pasta water 2 tablespoons at a time until it reaches a glossy, creamy consistency.

Stir in a small cold knob of butter just before serving to create a silky, restaurant-quality finish.