Preheat the oven to 350F. Shred the meat off your chicken (don't worry if some is left clinging to the bones), chop it and set aside. Place the carcass in the oven and roast for about half an hour; this will darken and enrichen the broth, but it isn't necessary. Transfer the carcass to a large pot with onion, one stalk of celery, water, garlic, ginger and peppercorns. Bring to a simmer and cook, covered, for about 45 minutes. Remove from heat and set aside until slightly cooled. Pour through a colander or sieve into another pot, and pull any chunks of chicken off the bones and add to the stock. Discard the rest.
Boil the noodles, barley or rice according to the package directions. Bring the broth to a simmer, add the other stalk of celery and the carrot, and cook for about 7 minutes, until they are tender. Add the cooked noodles, chopped chicken, and green onion. Season with salt and pepper and serve hot.