In a large stock pot, combine chicken bones with onion, carrot, celery, and peppercorns to create flavorful chicken stock.
Wrap the parsley, thyme, and bay leaves in a small piece of muslin, secure with kitchen twine, and add to the pot. Pour in the water and bring to a boil over high heat.
Reduce the heat to very low and delicately skim off any scum that rises to the surface using a slotted spoon. Simmer steadily for 2 hours, remembering that the longer you simmer, the richer the flavor of the stock will be.
Once off the heat, allow the mixture to cool for 1 hour. Line a fine sieve with a large piece of muslin, placing it over a bowl. Gently strain the stock using a ladle, discarding any solids.
In a large saucepan, combine chicken stock, soy sauces, wine, mandarin peel, cinnamon, star anise, sugar, and chicken breasts. Simmer until chicken is fully cooked, about 20 minutes. Remove chicken with a slotted spoon and let it cool on a plate for 10 minutes.
Shred the chicken using 2 forks. Add the shredded chicken back to the pan with the noodles and cook until noodles are tender, about 2 minutes. Stir in the carrot, snowpeas, shallots, and sesame oil until well combined. Remove from heat and serve.
Divide the soup into bowls, garnish with fresh coriander and mint leaves, and serve promptly.