Preheat oven to 375°F (190°C) and lightly grease a 9x13 inch baking dish.
Cook penne in a large pot of boiling salted water for 2 minutes less than the package instructions suggest.
Add broccoli florets to the boiling pasta water during the last 60 seconds of cooking, then drain both.
In a medium saucepan, melt butter over medium heat and whisk in flour, cooking for 1 minute to form a roux.
Gradually whisk in milk and garlic, stirring constantly until the sauce thickens and reaches a light simmer.
Remove sauce from heat and stir in parmesan and 1/2 cup of the mozzarella until melted; season with salt and pepper.
In a large mixing bowl, combine the drained pasta, broccoli, cubed chicken, and the cheese sauce.
Transfer the mixture to the prepared baking dish and sprinkle the remaining mozzarella over the top.
Bake for 20-25 minutes until the sauce is bubbly and the cheese is golden brown.