Cook the egg noodles in 1 quart boiling water until al dente, about 3 minutes. Drain and toss with a small amount of toasted sesame oil to prevent sticking. Set aside.
Have the stock heating in a pot.
Heat a large wok over high heat. When the wok begins to smoke, add the peanut oil. Stir-fry the ginger and chile for less than 30 seconds. Add the chicken and let settle for 1 minute before stir-frying.
As the chicken begins to brown, add the rice wine and stir-fry until the wine has almost evaporated. Carefully stir in the tofu. Then add in the mushrooms, bamboo shoots and the white parts of the napa cabbage, and stir-fry for 1 minute.
Add the hot stock. Bring to a boil, and then reduce to a simmer. Season the soup with the soy sauce, black rice vinegar, salt and ground white pepper. Stir in the green parts of the napa cabbage and continue to simmer for 10 minutes, skimming off any foam that arises.
Divide the noodles into 4 bowls. Ladle some of the soup over the noodles and garnish with scallions and cilantro.