Heat oven to 375°F. Lightly grease a 9x13-inch baking dish.
Warm butter and olive oil in a skillet over medium heat.
Soften the onion for 3 to 4 minutes, then stir in garlic for 30 seconds.
Stir in the uncooked rice and toast for 1 minute, stirring often.
Whisk broth, condensed soup, sour cream, thyme, paprika, salt, and pepper in a bowl.
Combine the rice mixture and frozen peas and carrots in the baking dish.
Pour the soup mixture over the rice and stir to coat evenly.
Nestle chicken pieces into the mixture. Cover the dish tightly with foil.
Bake for 35 minutes. Remove foil, stir gently, and sprinkle with cheddar.
Bake uncovered 12 to 15 minutes more, until rice is tender and chicken reaches 165°F.
Rest 5 to 10 minutes, then garnish with parsley and serve warm.