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.