Combine carcass, onion, carrot, celery, garlic, rosemary, thyme, bay leaves, salt, and pepper in a stockpot. Pour in enough water to cover; bring stock to a boil, cover the pot, reduce heat, and simmer until flavors blend, about 1 hour.

Remove meat from bones and set aside; discard carcass. Remove and discard vegetables and bay leaves with a slotted spoon.

Stir in onions, carrots, celery, garlic, poultry seasoning, rosemary, and onion powder. Bring soup to a boil; reduce heat, cover the pot, and simmer until vegetables are tender, 20 to 30 minutes.

Add cooked rice and reserved meat; season with salt and pepper. Cook until warmed through, about 5 minutes.