Heat oil in a skillet over medium heat. Add chicken, salt, and pepper; cook and stir until golden brown, about 5 minutes. Add garlic; cook until fragrant, about 2 minutes. Add broth and heavy cream; stir in pasta and bring to a boil. Reduce heat and simmer until pasta is tender, about 15 minutes.
Set oven rack about 6 inches from the heat source and preheat the oven's broiler. Grease an 11x17-inch baking pan.
Pour 1/2 the pasta mixture into the prepared pan. Sprinkle 1/2 the Parmesan cheese and 1/2 the mozzarella cheese over the pasta. Layer remaining pasta on top; finish with remaining Parmesan and mozzarella cheeses.
Broil in the preheated oven until cheese is golden brown, 10 to 15 minutes.