Preheat oven to 350°F (175°C). Lightly grease a 9x13-inch baking dish with butter or cooking spray.
In a large bowl, whisk together the condensed soup, chicken broth, sour cream, milk, garlic powder, onion powder, thyme, salt, and pepper until smooth.
Stir in the uncooked rice, shredded chicken, diced carrot, thawed peas, and 3/4 cup of the shredded cheddar cheese. Mix until everything is evenly coated.
Pour the mixture into the prepared baking dish and spread it into an even layer. Cover tightly with aluminum foil and bake for 25 minutes.
Remove foil, sprinkle the remaining 1/4 cup cheese over the top, and return to the oven uncovered for 5-10 minutes until bubbly and golden on top. Before serving, check that the casserole has reached an internal temperature of 165°F (74°C) in the center. Let stand 5 minutes before serving.