Cook Pasta and Broccoli: In a large pot, bring salted water to a boil. Add pasta and cook until al dente, about 8-10 minutes. Add broccoli florets during the last 3 minutes. Reserve 1 cup of pasta water, then drain.

Sauté Aromatics: In a skillet over medium heat, combine olive oil and butter. Add diced onion and sauté for about 4-5 minutes until translucent, then stir in minced garlic for 30 seconds.

Prepare Cream Sauce: Stir in heavy cream and chicken broth, then let simmer over low heat for 2-3 minutes.

Incorporate Cheeses: Remove from heat and whisk in Parmesan and mozzarella until melted. Add Italian seasoning, garlic powder, and red pepper flakes.

Combine Pasta and Chicken: Gently fold in the pasta, broccoli, and shredded chicken. Adjust sauce thickness with reserved pasta water.

Finish with Butter: Stir in a cold knob of butter before serving for a silky finish.