Preheat the oven to 350°F (175°C). Lightly grease a 9×13 inch baking dish.
Bring a pot of water to a boil and blanch the broccoli florets for 2 minutes until bright green and slightly tender. Drain and set aside.
Heat olive oil in a medium skillet over medium heat. Add chopped onion and minced garlic and sauté for 3-4 minutes until softened and fragrant.
In a large mixing bowl, combine cooked rice, shredded chicken, blanched broccoli, sautéed onion and garlic, Greek yogurt, chicken broth, dried thyme, salt, black pepper, and paprika. Stir until well mixed.
Fold in 1/2 cup shredded cheddar cheese and all of the mozzarella cheese into the mixture.
Transfer the mixture into the prepared baking dish and spread evenly.
Sprinkle the remaining 1/2 cup shredded cheddar cheese evenly over the top.
Bake in the preheated oven for 30-35 minutes until the casserole is hot and bubbly and the cheese on top is melted and slightly golden.
Remove from oven and let cool for 5 minutes before serving.