Place turkey carcass and water in a large Dutch oven. May add additional celery, carrots and onions not listed in recipe. Bring to a boil. Cover, reduce heat, and simmer 1 hour. Remove carcass from broth, and remove meat from bones. Set broth and meat aside. Measure broth; add water, if necessary, to measure 3 quarts
Heat butter in a large Dutch oven; add flour, and cook over medium heat 5 minutes, stirring constantly. (Roux will be a blond color.)
Add onion, carrot, and celery to roux; cook over medium heat 10 minutes, stirring often. Add 3 quarts broth, turkey meat, rice, salt, and pepper; bring to a boil. Cover, reduce heat, and simmer 20 minutes or until rice is tender. Add half-and-half; cook until thoroughly heated.