Preheat your oven to 375°F (190°C) and grease a 2-quart casserole dish with a light coating of butter.
In a skillet, sauté the chopped onion in a touch of oil or butter over medium heat until translucent, about 3–4 minutes. Add the minced garlic and cook for another 30 seconds, just until fragrant.
In a large bowl, combine the rice, chicken broth, sour cream, shredded chicken, vegetables, thyme, salt, and pepper. Stir until everything is evenly coated and distributed.
Pour the mixture into the prepared baking dish. Dot the top with the remaining butter and sprinkle with half of the shredded cheddar.
Cover the dish with foil and bake for 35 minutes. Remove the foil, give the casserole a gentle stir to ensure even cooking, then sprinkle the remaining cheddar on top.
Return to the oven, uncovered, and bake for an additional 10–15 minutes, until the rice is tender and the cheese is melted and bubbling.
Let the casserole rest for about 5–10 minutes before serving so the flavors set and the dish firms up slightly for easy scooping.