Preheat the oven to 350°F (175°C).
Steam the broccoli florets until tender but still crisp, about 4-5 minutes. Drain and set aside.
In a large skillet, melt the butter over medium heat. Add the chopped onion and minced garlic, sauté until softened and fragrant, about 3-4 minutes.
In a large mixing bowl, combine the cooked rice, shredded chicken, steamed broccoli, sautéed onion and garlic, sour cream, condensed cream of chicken soup, milk, black pepper, and salt. Stir well to combine.
Add the entire packet of Knorr Chicken Flavor Rice Sides mix (dry seasoning only, do not prepare separately) into the mixture and stir until evenly distributed.
Fold in 1 cup of shredded cheddar cheese and 1/2 cup of shredded mozzarella cheese into the mixture.
Transfer the mixture into a greased 9×13 inch casserole dish and spread evenly.
Sprinkle the remaining 1/2 cup cheddar cheese and 1/2 cup mozzarella cheese evenly over the top of the casserole.
Bake uncovered in the preheated oven for 35-40 minutes, or until the casserole is bubbly and the cheese on top is melted and golden.
Remove from the oven and let it rest for 5 minutes before serving.