Heat oil in a large pot or Dutch oven over medium-high; season chicken with salt and pepper and cook 5-6 minutes until golden brown and cooked through. Set aside.
Cook onion in same pot 3-4 minutes until softened, then garlic another minute.
Add broth, cream, Italian seasoning, and optional pepper flakes; bring to a boil.
Stir in pasta; reduce heat to medium and cook 8-10 minutes, stirring occasionally, until almost tender.
Add broccoli and return chicken; cook 4-5 minutes until broccoli is tender-crisp and pasta fully cooked.
Off heat, stir in mozzarella and Parmesan until melted and creamy; adjust salt and pepper and stand 2-3 minutes to thicken sauce.