Begin by stripping the meat from the carcass. Set the meat aside until needed. Place turkey carcass in a soup kettle or Dutch oven and cover with enough water to fully immerse. Bring to a boil, then reduce heat to low. Cover and simmer for one hour or longer. Remove carcass and strain 10 cups of turkey broth.
Remove remaining turkey meat from the carcass and slice into bite-size pieces. Add more turkey, as necessary, to equal four cups.
Melt butter in a large soup kettle or Dutch oven. Saute celery, carrots, onions, and garlic for a couple of minutes. Add flour, and cook for a few more minutes.
Gradually add half of the reserved broth, stirring until smooth. Bring to a boil, then simmer until thickened.
Add the remaining broth, barley, corn, salt, pepper, reserved turkey, and thyme. Reduce heat, cover, and simmer for 35 minutes or until barley is tender.
Add milk, and half-n-half, then simmer for an additional 10 minutes.