Preheat the oven to 375°F. Lightly grease a 9×13 baking dish and set it aside.
In a large bowl, combine the cooked rice, chicken, broccoli, and shredded cheese.
Add the sour cream, gluten free cream of chicken soup, egg, salt, and black pepper. Stir until evenly mixed.
Transfer the mixture to the prepared baking dish and spread it into an even layer.
Cover loosely with foil and bake for 30 minutes.
Remove the foil and bake for another 10 minutes until hot and lightly golden on top.
Remove from the oven and let the casserole rest for 10 minutes before serving.