Preheat the oven to 375°F (190°C).
In a medium saucepan, bring the chicken broth to a boil. Add the brown rice, reduce heat to low, cover, and simmer for 40 minutes or until the rice is tender and liquid is absorbed.
While the rice cooks, heat olive oil in a large skillet over medium heat. Add the chopped onion and cook for 3-4 minutes until softened.
Add the minced garlic and chicken pieces to the skillet. Cook, stirring occasionally, until the chicken is no longer pink and cooked through, about 6-8 minutes.
Add the broccoli florets to the skillet and cook for an additional 3-4 minutes until they are bright green and slightly tender.
In a large mixing bowl, combine the cooked rice, chicken and broccoli mixture, cream of mushroom soup, Greek yogurt, shredded cheddar cheese, dried thyme, salt, pepper, and paprika. Stir until well combined.
Transfer the mixture to a greased 9×13-inch baking dish and spread evenly.
Sprinkle the shredded mozzarella cheese evenly over the top of the casserole.
Bake in the preheated oven for 20 minutes or until the casserole is bubbly and the cheese is melted and golden.
Remove from the oven and let it rest for 5 minutes before serving.