How to Make It
Using a sharp knife, remove the breast meat from the chicken. Put the rest of the chicken in a large pot and add the sliced ginger, peppercorns and water. Bring to a boil over moderately high heat, skimming. Add 1 tablespoon of salt, reduce the heat to low and simmer for 1 1/2 hours, skimming occasionally.
Meanwhile, bring a large saucepan of salted water to a boil. Skin the chicken breasts and cut the meat into 1-inch pieces. Put the meat in a food processor, add 1 tablespoon of the fish sauce, the cornstarch, vegetable oil and sugar and process to a fine paste. Scrape the paste into a large bowl and stir in the mushrooms and scallion.
Using 2 teaspoons, shape the chicken mixture into 24 dumplings. Add the dumplings to the boiling water and simmer over moderate heat until firm and springy, about 5 minutes. Using a slotted spoon, transfer the dumplings to a bowl and cover with a kitchen towel. Reserve the cooking liquid.
Put the rice noodles in a large bowl, cover with cold water and let soak until pliable, about 25 minutes. Drain and return to the bowl.
Strain the chicken stock. Remove the meat from the chicken, discarding the skin. Tear the chicken meat into large pieces. Return the chicken stock to the pot, add the remaining 1/4 cup of fish sauce and season with salt and pepper. Bring to a gentle simmer and keep warm.
Bring the dumpling cooking liquid back to a boil. For each serving, put one quarter of the rice noodles in a small strainer and cook them in the boiling liquid, stirring occasionally, until al dente, about 40 seconds. Lift out the strainer, shaking the noodles to drain them, and turn them out into a soup bowl.Add one quarter of the chicken meat and dumplings to the strainer and rewarm in the cooking liquid, then add to the rice noodles. Ladle about 3 cups of the chicken broth on top and garnish with snow peas, cilantro and a few drops each of sesame oil and chili oil. Top with bean sprouts and serve at once. Repeat with the remaining ingredients.