In a large stockpot, combine the meaty leftover turkey carcass, chicken broth, and water. Add chopped celery, carrot, poultry seasoning, bay leaf, salt, and pepper. Bring to a vigorous boil, then simmer for 3 hours.

Remove the turkey carcass, cool slightly, shred the turkey meat, and set aside 4 cups for the soup.

Strain the broth through a fine mesh sieve, discarding the cooked vegetables and bay leaf. Return the broth to the stockpot.

Add the chopped onion, celery, carrot, and green beans to the strained broth. Bring to a boil and simmer for 10 minutes until the vegetables are tender.

Stir in the frozen corn, peas, and reserved shredded turkey. Bring back to a gentle boil for 5 minutes.

In a separate bowl, mix the biscuit mix with milk until just combined. Drop spoonfuls into the simmering broth and cover for 10 minutes without lifting the lid.

Check the dumplings for doneness, then ladle the soup into bowls and serve.