Combine the chicken stock, water, sesame oil, Tamari or soy sauce and ginger in a large soup kettle. Cut the lime in half, squeeze the juice into the mixture and toss in the rinds. Bring to a boil. Add the chicken thighs, reduce heat to a simmer and poach the chicken thighs for 12 – 13 minutes or until cooked through. Remove from the broth along with the lime rinds. Discard the limes.
Raise the heat and bring the stock back up to a full boil, add the pasta and cook for 6 – 8 minutes, stirring occasionally.
While the pasta is cooking, shred the chicken meat using 2 forks. Cover and keep warm.
When the pasta is done, remove from the stock with tongs or a strainer and divide among four soup bowls. Add the peas to the boiling stock and cook for 2 minutes. Remove with a strainer or slotted spoon and set aside, cook the carrots in the same manner for 1 minute and the mushrooms for 30 seconds. (I run all the vegetables individually under hot water in a strainer and place on a plate until I am ready to finish off the soup bowls.)
Taste the broth and add salt and pepper if needed. Ladle the broth over the noodles and arrange the chicken and veggies in individual little piles on top. Garnish with a sprinkling of sesame seeds and herb leaves if desired and serve right away.