Preheat the oven to 350°F (175°C). Lightly grease a 9×13 inch casserole dish.
In a medium saucepan, melt the butter over medium heat. Add the chopped onion and minced garlic and sauté for 3-4 minutes until softened and fragrant.
Add the chicken broth and milk to the saucepan and bring to a simmer.
Stir in the instant rice, salt, black pepper, and paprika. Remove from heat, cover, and let the rice absorb the liquid for 5 minutes.
Meanwhile, steam the broccoli florets until just tender, about 4-5 minutes. Drain any excess water.
In a large mixing bowl, combine the cooked chicken, steamed broccoli, sour cream, shredded cheddar cheese, shredded mozzarella cheese, and the cooked rice mixture. Stir until evenly combined.
Transfer the mixture to the prepared casserole dish and spread evenly.
Bake uncovered in the preheated oven for 20-25 minutes, until the casserole is bubbly and the cheese is melted and slightly golden on top.
Remove from the oven and let cool for 5 minutes before serving.