Preheat the oven to 350°F (175°C).
In a large skillet, heat the olive oil over medium heat. Add the sliced onions and cook, stirring frequently, until caramelized and golden brown, about 30 minutes.
Add the minced garlic, dried thyme, salt, and black pepper to the onions. Stir and cook for 1 more minute. Remove from heat and set aside.
In a medium saucepan, melt the butter over medium heat. Stir in the flour and cook for 1-2 minutes until lightly browned, stirring constantly.
Slowly whisk in the chicken broth and white wine. Bring to a simmer and cook until slightly thickened, about 5 minutes.
In a large mixing bowl, combine the uncooked rice, chicken pieces, caramelized onion mixture (reserve 1/4 cup for topping), and the broth mixture. Stir well to combine.
Transfer the mixture to a greased 9×13 inch casserole dish and spread evenly.
Cover the casserole dish tightly with aluminum foil and bake for 50 minutes.
Remove the foil and sprinkle the shredded Gruyère and mozzarella cheeses evenly over the top along with the reserved caramelized onions.
Return the casserole to the oven, uncovered, and bake for an additional 15 minutes, or until the cheese is melted and bubbly.
Remove from the oven and let rest for 5 minutes before serving.