Put the chicken broth, turkey, carrot halves, celery stalk, onion halves and 1 bay leaf in a large stockpot. Bring to a boil and then simmer, about 1 1/2 hours.
Dice the turkey meat. Make sure the meat pieces are no larger than the size of a soupspoon. (If preparing the soup the next day, be sure to store leftover turkey meat in an airtight container before placing it in the refrigerator. Top with 1 or 2 ladles full of broth to keep the meat moist.)
Before straining the broth, remove the large bones and carcass with tongs. Strain the broth through a sieve covered with wet cheesecloth. Discard the solids. Transfer the broth to a bowl set in a bath of ice water, which will cool the broth quickly and help keep it fresher longer. This can be done the night before and stored in the refrigerator until the next day.
In a large soup pot, heat the garlic in the olive oil over medium heat. Allow to brown slightly, about 3 minutes. Add the minced carrots, celery and onions. Sweat over medium-low heat until softened, 7 or 8 minutes.
Dice the leftover Thanksgiving vegetables. Add the sage to the soup pot along with the turkey broth and the remaining bay leaf. Bring to a simmer. When simmering, add the Brussels sprouts, green beans and diced turkey meat to the soup. Bring it back up to a simmer. Finally, add the sweet potatoes to the center and gently push them down. Turn the heat off and cover. Allow to sit and steam, 5 to 7 minutes. Let simmer for 5 more minutes and serve with the Mashed Potato Polpetti.
Re-season the potatoes with salt and pepper. Stir in the cheese. Take a mayonnaise or peanut butter lid and line with plastic wrap to use as a mold. Pack the potato mixture into the lid, unwrap it and place the patty on a baking sheet. You may refrigerate them covered with plastic wrap until the next day, or fry immediately.
To cook, dredge the patties in flour. Heat the oil in a nonstick skillet over medium-high heat. Just before frying, dredge the patties again with flour.
Add the patties to the oil without crowding them (you may have to cook them in batches). Cook until the underside is golden brown, about 5 minutes. Flip and brown the other side. Remove from the skillet and drain on paper towels. Serve hot with leftover gravy. Makes 6 servings.
From Food Network Kitchens; after further testing and to ensure the best results this recipe has been altered from what was in the actual episode.