In a small saucepan, bring 3 cups chicken stock to a boil. Add chicken breast; poach until cooked through, about 12 minutes. Transfer chicken to a cutting board; slice in half crosswise, then into 1/4-inch-thick strips. Set aside. Pass stock through a cheesecloth-lined sieve set over a medium bowl; set aside. In a 6-quart saucepan, heat sesame oil over medium-low heat. Add garlic and onion, and cover. Saute, stirring occasionally, until translucent, 6 to 7 minutes. Add mushrooms, and saute until tender, about 5 minutes. Add reserved and remaining chicken stock, carrots, ginger, black pepper, cayenne, and salt; bring liquid to a boil. Reduce the heat, and simmer until carrots are tender, about 3 minutes.
Place cornstarch in a small bowl, and slowly whisk in soy sauce until cornstarch has dissolved. Whisk in vinegar. Slowly whisk cornstarch mixture into soup. Simmer until soup is clear, 5 to 7 minutes. Add reserved chicken, and continue simmering until chicken is heated through, about 2 minutes. Turn off the heat. Slowly pour beaten eggs through a perforated spoon into soup. Cover soup, and let stand 1 minute. Remove the cover, and gently stir. Add scallions and tofu. To serve, ladle into individual bowls.
Place peppercorns, dill, parsley, bay leaves, leeks, carrots, celery, chicken, wings, and backs into a large stockpot. Add stock and 6 cups cold water. Bring to a boil. Reduce to a very gentle simmer, and cook for 45 minutes. Liquid should just bubble up to the surface. A skin will form on the surface of the liquid; skim this off with a slotted spoon, and discard. Repeat as needed. After 45 minutes, remove chicken from the pot, and set aside until it is cool enough to handle. Remove the meat from the bones, set the meat aside, and return the bones to the pot. Shred the chicken, and set aside in the refrigerator until ready to use. Continue to simmer the stock, on the lowest heat possible, for 3 hours, skimming as needed. The chicken bones will begin to disintegrate. Strain the stock through a fine sieve into a very large bowl. Discard the solids. Place the bowl in an ice bath, and let cool to room temperature. Transfer to airtight containers. Stock may be refrigerated for 3 days or frozen for 4 months. Refrigerate for at least 8 hours, or overnight. If storing, leave fat layer intact to the seal the stock. Before using, remove the layer of fat that has collected on the surface.
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