Preheat the oven to 375°F (190°C).
In a medium saucepan, bring the chicken broth to a boil. Add the rice, reduce heat to low, cover, and simmer for 18 minutes or until the rice is tender and liquid is absorbed. Remove from heat and set aside.
While the rice cooks, heat olive oil in a large skillet over medium heat. Add the chopped onion and cook for 4-5 minutes until softened.
Add the minced garlic to the skillet and cook for 1 minute until fragrant. Remove from heat.
In a separate medium saucepan, melt the butter over medium heat. Whisk in the flour and cook for 1-2 minutes to form a roux, stirring constantly.
Gradually whisk in the milk, continuing to whisk until the sauce thickens, about 3-4 minutes.
Remove the sauce from heat and stir in the shredded cheddar cheese, mozzarella cheese, salt, pepper, dried thyme, and paprika until smooth and fully melted.
In a large mixing bowl, combine the cooked rice, cooked onion and garlic mixture, shredded chicken, and cheese sauce. Stir until well combined.
Transfer the mixture to a greased 9×13 inch baking dish. Sprinkle the grated Parmesan cheese evenly over the top.
Bake uncovered in the preheated oven for 20-25 minutes until the casserole is bubbly and the top is golden brown.
Remove from the oven and let rest for 5 minutes before serving.