In a large stockpot, combine the turkey carcass, chicken broth, and water with chopped celery, sliced carrot, poultry seasoning, bay leaf, salt, and pepper. Bring to a vigorous boil.

Once boiling, reduce heat to low and cover. Simmer for 3 hours, stirring occasionally.

Carefully remove the turkey carcass from the pot and let cool. Pick the meat from the bones, aiming for about 4 cups. Return meat to the pot.

Strain the broth through a fine-mesh sieve and return the liquid to the stockpot. Stir in chopped onion, additional celery, diced carrots, and green beans.

Reduce heat and simmer for about 10 minutes until vegetables are tender.

Mix in frozen corn, peas, and the reserved turkey. Bring back to a gentle boil.

In a separate bowl, mix the biscuit mix with milk until a soft dough forms. Drop spoonfuls of the dough into the soup.

Cover and let the dumplings simmer for 10 minutes without lifting the lid.