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 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.