Bring the stock to a boil over medium heat in a soup pot. Add the lime leaves, chiles, garlic, ginger and lemongrass. Lower the heat to medium-low; cover the pot and gently simmer to let the spices infuse the broth, about 10 minutes.
Uncover the pot and stir in the chicken, coconut milk, mushrooms, fish sauce, sugar and lime juice. Simmer to heat the chicken through, about 5 minutes. Sprinkle with salt and pepper. Ladle the soup into a soup tureen or individual serving bowls. Garnish with the cilantro. Be careful to avoid chewing the lemongrass, ginger or lime leaves.
Place the chicken and vegetables in a large stockpot over medium heat. Pour in only enough cold water to cover (about 3 quarts); too much will make the broth taste weak. Toss in the thyme, bay leaves, and peppercorns, and allow it to slowly come to a boil. Lower the heat to medium-low and gently simmer for 1 to 1 1/2 hours, partially covered, until the chicken is done. As it cooks, skim any impurities that rise to the surface; add a little more water if necessary to keep the chicken covered while simmering.
Carefully remove the chicken to a cutting board. When its cool enough to handle, discard the skin and bones; hand-shred the meat into a storage container.
Carefully strain the stock through a fine sieve into another pot to remove the vegetable solids. Use the stock immediately or if you plan on storing it, place the pot in a sink full of ice water and stir to cool down the stock. Cover and refrigerate for up to one week or freeze.
Cook Time: 1 hour, 30 minutes
Yield: 2 quarts stock
Lemongrass imparts a delicate citrusy flavor and aroma, but if you can't find lemongrass, use equal parts lemon peel and ginger. Kaffir lime is gently limey with lots of aroma, but you can substitute lime peel if you can't find it.
This recipe has been updated and may differ from what was originally published or broadcast.