Dice the carrot, green onion, and leek into small pieces. Slice the shiitake mushrooms.

In a large pot, heat about 2 tablespoons of sesame oil. Add the carrot and cook until it starts to brown. Add the green onion and leek, sauté for a few minutes, then add the shiitake mushrooms.

Add the rice vinegar, stir, and let it evaporate. Pour in the chicken broth, add the 2 dried chili peppers, black pepper, ginger, and soy sauce. Bring to a boil and simmer for about 20 minutes.

While the soup simmers, cut the chicken breast into small cubes. In a separate bowl, mix the cornstarch with the water.

After 20 minutes, add the chicken breast to the soup and cook for a few minutes. Then add the cornstarch mixture, stir, and let the soup thicken slightly.

Beat the egg as you would for scrambled eggs and slowly pour it into the soup while stirring constantly. Serve very hot.