Warm the butter and olive oil in a large pot over medium heat. Add the onion, carrots, celery, and garlic. Cook for 5 to 6 minutes, stirring now and then, until the vegetables soften and smell fragrant.

Sprinkle the flour over the vegetables and stir well. Cook for 1 to 2 minutes so the flour loses its raw taste. Keep stirring to coat the vegetables evenly.

Pour in the stock a little at a time, stirring as you go to prevent lumps. Add the turkey and any bones, along with the poultry seasoning, salt, and pepper. Bring the soup to a boil, then lower the heat and simmer gently for 15 to 20 minutes.

While the soup simmers, mix the stuffing, egg, and flour in a bowl. Stir only until the mixture holds together. Shape it into balls about the size of a heaping tablespoon, handling the mixture gently so the dumplings stay tender.

Gently drop the dumplings into the simmering soup. Cover the pot and cook for 10 to 12 minutes, until the dumplings are cooked through and floating. Keep the soup at a gentle simmer rather than a rolling boil.

Remove any turkey bones from the pot, checking the soup carefully for small pieces. Taste the broth and adjust the salt and pepper as needed. Serve the soup hot.