In a large saucepan set over moderate heat, heat the oil until it is hot. Add the onion, garlic, and ginger, and cook, stirring occasionally, 3 minutes. Add the carrot and cabbage and toss to combine Add the broth, reserved mushrooms and salt to taste and simmer 10 minutes. In a saucepan of boiling salted water cook the pasta until al dente, drain and transfer to stock. Add the broccoli, chicken and seasonings and simmer stirring occasionally, for 3 minutes, or until broccoli is just cooked and chicken heated through. Add sesame oil. Ladle the soup into bowls and garnish with the scallion.