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

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

In a large skillet over medium-low heat, heat the olive oil and butter together. Add the diced onion and cook for 4 to 5 minutes until soft and translucent. Add the minced garlic and cook for 30 seconds until fragrant. Do not let the garlic brown.

Pour in the heavy cream and chicken broth. Bring to a gentle simmer for 2 to 3 minutes, stirring occasionally. Look for small bubbles at the edges, not a rolling boil.

Remove the skillet completely from heat. Whisk in the Parmesan and mozzarella until the sauce is totally smooth. Season with Italian seasoning, garlic powder, salt, pepper, and red pepper flakes if using.

Add the drained pasta and broccoli to the sauce and toss gently with tongs until evenly coated. Fold in the shredded rotisserie chicken last to keep it tender. Add the reserved pasta water 2 tablespoons at a time until the sauce is creamy and glossy. Stir in a small cold knob of butter just before serving for a silky, restaurant-quality finish.