Preheat oven to 350 degrees F (175 degrees C). Lightly oil a 9x13-inch baking dish.
Place carrots in a microwave-safe bowl. Add 1 tablespoon water. Cover tightly with plastic wrap; poke holes in wrap to vent. Microwave on High for 1 minute. Drain.
Combine chicken, rice and contents of seasoning packet, carrots, mushrooms, onion, peas, and spinach in a bowl. Stir in broth and soup. Pour mixture into prepared baking dish. Cover with foil.
Bake, covered, 45 minutes. Remove foil and bake until hot and bubbling, about 45 more minutes. Top with shredded cheese just before casserole is done.
