There's probably no better way to empty that overstuffed post-Thanksgiving refrigerator than a hearty soup. Using the turkey bones to make a broth for the soup is the best way to get every penny's worth from your bird. If your leftover stuffing is chunky and rustic, finely chop it to make these delicate dumplings.
Put the turkey carcass, quartered onions, coarsely chopped carrots and celery, smashed garlic, bay leaf, and peppercorns in a large stockpot and add enough cold water to just cover, about 2 quarts. Bring the water to a boil, and then reduce the heat to maintain a gentle simmer and cook for 1 hour. Remove from the heat and strain the solids from the broth. Pour the liquid through a fine mesh strainer and reserve; you should have about 10 to 12 cups broth.
Meanwhile, in a small bowl, whisk the eggs, flour, 1/2 teaspoon salt, and some black pepper together until smooth. Add the stuffing and mix until well combined; cover and reserve.
Wipe the stockpot clean with a paper towel. Heat the oil in the pot over medium heat. Add the chopped onion and garlic and cook until soft and translucent, about 6 minutes. Add the sliced carrots and celery, thyme sprigs, and reserved broth and bring to a simmer; cook vegetables are just soft, about 10 minutes.
Roll level tablespoons of the dumpling mixture into balls with wet hands (see note) and drop into the simmering soup; cook until dumplings float, 3-4 minutes. Gently stir in the turkey meat, corn, and season with salt and pepper, and simmer until heated through. Serve immediately.
Moistness of stuffing can vary; if the dumpling dough is too soft to roll, add flour a teaspoon at a time until it is firm enough to hold its shape while rolling.