Preheat the oven to 350°F (175°C).
Bring a pot of water to a boil and blanch the broccoli florets for 2-3 minutes until bright green and slightly tender. Drain and set aside.
Heat olive oil in a skillet over medium heat. Add the chopped onion and minced garlic, sautéing for 3-4 minutes until softened and fragrant.
In a large mixing bowl, combine the cooked rice, cooked chicken, blanched broccoli, sautéed onion and garlic, cream of mushroom soup, sour cream, milk, salt, black pepper, paprika, and dried thyme. Stir well to combine all ingredients evenly.
Fold in 3/4 cup of shredded cheddar cheese into the mixture.
Transfer the mixture into a greased 9×13 inch casserole dish and spread evenly.
Sprinkle the remaining 1/4 cup 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 on top is melted and slightly golden.
Remove from the oven and let the casserole rest for 5 minutes before serving.