Heat olive oil in a large pot or Dutch oven over medium-high heat. Season chicken pieces with salt and pepper, then add to the pot. Cook for 5-6 minutes until golden brown and cooked through. Remove chicken and set aside.
In the same pot, add the diced onion and cook for 3-4 minutes until softened. Add garlic and cook for another minute.
Pour in the chicken broth and heavy cream, then add Italian seasoning and red pepper flakes if using. Bring to a boil.
Add the pasta and stir well. Reduce heat to medium and cook, stirring occasionally, for 8-10 minutes until pasta is almost tender.
Add the broccoli florets and return the cooked chicken to the pot. Continue cooking for 4-5 minutes until broccoli is tender-crisp and pasta is fully cooked.
Remove from heat and stir in the mozzarella and Parmesan cheeses until melted and creamy. Season with additional salt and pepper if needed. Let stand for 2-3 minutes before serving to allow the sauce to thicken.