Put turkey carcass, quartered onions, chopped carrots and celery, smashed garlic, bay leaf and peppercorns in a large stockpot and add just enough cold water to cover the carcass, which should be about 2 quarts. Bring to boil and then reduce heat and maintain a simmer for 1 hour. Remove from heat and strain the solids. Pour the liquid through a fine mesh strainer and reserve the stock which should give you about 10 to 12 cups.
In large bowl, whisk together the eggs, flour, salt and black pepper until smooth. Add the stuffing and mix until combined. Set aside. (If stuffing is still too soft to roll, add flour a tablespoon at a time until it is firm enough to hold its shape when rolling)
In same stockpot that had cooked stock, heat oil and butter over medium heat. Add onion and cook until soft and translucent, about 6 minutes. Add carrots, celery, thyme, sage, rosemary and garlic and cook for another 5 minutes. Add reserved stock and bring to simmer. Cook vegetables until they are soft, about 10 minutes.
Use wet hands and roll dumping mixture into balls to about tablespoon size and drop into the soup. (The dumplings do not need to be a perfect size or shape). Add all of the dumplings and cook for about 3-4 minutes, until the dumpling float.
Add turkey and corn and simmer until heated through. Taste the soup and adjust with salt and pepper.
Use quality low-sodium turkey stock as it contributes to the flavor. You can also use chicken stock which works for the soup as well.
If stuffing dumplings are too soft to roll into balls, add 1 tablespoon of flour at a time until you can easily roll them.
If you have other leftover vegetables from Thanksgiving, feel free to add them to the dumplings or the soup.
Use fresh herbs if possible, but if you don't have fresh herbs, reduce the amount by ⅔rds for dried herbs as they are much more potent.