Preheat the oven to 375°F (190°C).
Cook the pasta according to the package instructions until al dente. Drain and set aside.
In a large skillet, heat the olive oil over medium-high heat. Add the chicken and cook until browned and cooked through, about 5-6 minutes. Remove the chicken from the skillet and set aside.
In the same skillet, add the diced onion and cook until softened, about 3-4 minutes. Add the minced garlic and cook for an additional minute.
Add the broccoli to the skillet and cook until tender, about 3-4 minutes.
In a large bowl, whisk together the chicken broth, heavy cream, thyme, salt, and pepper. Stir in the cooked pasta, chicken, and broccoli mixture.
In a greased 9x13-inch baking dish, arrange half of the pasta mixture in the bottom of the dish. Top with half of the shredded cheddar cheese and half of the grated Parmesan cheese.
Repeat the layers, starting with the remaining pasta mixture, then the remaining cheddar cheese, and finally the remaining Parmesan cheese.
Cover the dish with aluminum foil and bake for 25-30 minutes, or until the cheese is melted and bubbly
Remove the foil and bake for an additional 5-10 minutes, or until the top is lightly browned.
Serve hot and enjoy!