Cook pasta according to package directions. Drain and set aside.
In a large skillet, heat olive oil over medium-high heat.
Season chicken with salt and pepper, then add to the skillet and cook until browned and cooked through. Remove from the skillet and set aside.
Add minced garlic and broccoli to the same skillet. Sauté for 2–3 minutes.
Pour in chicken broth and heavy cream. Bring to a simmer.
Add the cooked pasta and chicken back into the skillet. Stir to combine.
Reduce heat and stir in mozzarella and Parmesan cheese until melted and creamy.
Serve warm with extra cheese on top, if desired.