Bring a medium saucepan of water to a boil. Add the noodles and cook until al dente. Drain and rinse under cold water.
Meanwhile, in a large saucepan, heat the oil. Add the ginger and cook over high heat until fragrant, about 1 minute. Add the chicken stock and carrot and bring to a boil. Add the chicken, cabbage, snow peas and scallion and simmer until the chicken is cooked through, about 5 minutes. Add the noodles and simmer until heated through. Serve in deep bowls, passing the soy sauce, hot sauce and sesame oil at the table.