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 medium skillet, melt the butter over medium heat. Add the chopped onion and cook until soft and translucent, about 3-4 minutes.
Add the minced garlic to the skillet and cook for 1 minute more until fragrant. Remove from heat.
In a large mixing bowl, combine the cooked rice, shredded chicken, steamed broccoli, cooked onion and garlic, cream of mushroom soup, sour cream, milk, salt, black pepper, and dried thyme. Stir until all ingredients are well incorporated.
Transfer the mixture to a greased 9×13-inch baking dish and spread evenly.
Sprinkle 3/4 cup of shredded cheddar 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 is melted and lightly golden.
Remove the casserole from the oven and sprinkle the remaining 1/4 cup shredded cheddar cheese on top. Let it rest for 5 minutes before serving.