For the carrots and leeks: Bring a saucepan of water to a boil and add some salt. Add the carrots to the water and simmer until tender; remove with a slotted spoon and set aside. Add the leeks to the water and simmer until tender. Drain completely. Refrigerate the leek and carrots until chilled.
For the chicken stock: Place the chicken and beef bone in a 6-quart stockpot and fill with 16 cups (4 quarts) of cold water. Bring to a boil, and then simmer, skimming any foam that forms on the surface, for 15 minutes. Add the celery leaves, carrots, leeks, onions, parsley and some salt; bring back up to a boil and then simmer, covered, for 45 minutes. Strain through a fine-mesh strainer until the stock is clear. Reserve the chicken for another use and discard the vegetables.
For the meatballs: Bring 8 cups (2 quarts) of salted water to a simmer in a shallow pan.
Mix together the pork, veal, 1 teaspoon salt, the white pepper, egg whites and 1/4 to 1/2 cup of the chicken stock in a bowl until well combined. Using a teaspoon, scoop some of the meatball mixture out and roll off the palm of your hand into the simmering water. Dip the spoon into hot water in between each meatball so the mixture does not stick to the spoon. Drop as many meatballs into the water to form a single layer. Cook for 4 to 5 minutes, then remove with a slotted spoon and let cool. Continue with the remaining meatball mixture. Set aside to cool completely.
For the dumplings: In a 2-quart Dutch oven over high heat, add the butter and 1 cup water and bring to a boil. Lower the heat and add the flour and salt. Stir together until a smooth dough forms, then remove from the pan into a bowl and let cool completely, about 30 minutes.
Bring 8 cups (2 quarts) of the chicken stock to a simmer in a shallow pan. Fill a bowl with ice water and set aside.
Meanwhile, in the bowl of a stand mixer fitted with a dough hook attachment, add the dough and begin kneading on low speed. Add the eggs one at a time, waiting for each one to be completely incorporated before adding the next. Take a resealable plastic bag and cut one corner off so you have a 1/2-inch opening. Fill the bag with half of the dough. While pressing the dough gently over the pot of simmering stock, begin cutting the dough as it comes out of the bag every 1 inch, using a dull knife or teaspoon dipped in the hot stock. Simmer for 4 to 5 minutes; do not let the stock boil. Using a slotted spoon, remove the dumplings from the pot and drop them immediately into the bowl of ice water to cool them down. Drain and lay out on a baking tray. Repeat this process with the remaining dough. Strain the chicken stock and reserve for another use.
Bring the remaining 8 cups (2 quarts) of chicken stock and the meatballs to a boil, lower the heat and simmer until the meatballs are heated through, 5 to 6 minutes. Remove from the heat and add the dumplings and chilled diced carrots. Ladle the soup into bowls and top with a teaspoon of the chilled leeks.
A viewer or guest of the show, who may not be a professional cook, provided this recipe. It has not been tested for home use.