Pour boiling water over noodles; leave 10 minutes. Wipe mushrooms with paper towel, remove stalks and slice large caps in half, leaving others whole.
Slice cabbage into quarters and remove core. Slice cabbage thinly. Cut carrots into fine strips. Drain noodles and cut into short lengths.
Combine noodles, cabbage and carrots in a bowl. Heat oil in a pan and sauté mushrooms over high heat. Add garlic and ginger and cook another 3 minutes. Stir in soy sauce and 1 tablespoon rice vinegar. Remove from heat. Add brown sugar, lime juice and salt and pepper to taste. Leave to cool.
Combine mushroom mixture and noodle mixture. Add remaining 1 tablespoon rice vinegar and season to taste with salt and pepper. Distribute among individual serving dishes. Sprinkle with chilies.