Place the chicken stock in a heavy-bottomed pot over high heat. Add the lime peel, lemon peel, ginger root, green chiles, and the cilantro stems and roots. Bring to a boil, turn down to a simmer and cook until reduced to 3 cups. Press occasionally on the solids with a wooden spoon to release flavor. Working with a sharp knife, cut the chicken breasts into broad slices, as thin as possible. Salt lightly, and reserve in refrigerator. When broth is reduced, strain into a measuring cup. Press on the solids in the strainer with a wooden spoon. Place the coconut milk in the soup pot over medium heat. Slowly stir in the reduced broth. Warm rapidly but do not allow to boil (otherwise the coconut milk will separate). When the soup is just below the boiling point, add the chicken slices. Remove from heat after 30 seconds (or when the chicken is just cooked through). Just before serving, mix together the lime juice and sugar in a small bowl. Blend in the fish sauce. Pour mixture into hot soup, season to taste and garnish soup with a little bit of chopped scallion and cilantro leaves. Yield: 6 first-course servings.
Optional step: If your soup has curdled a bit, or if you'd like a fluffier texture in the soup, before you add the chicken, place the soup in a blender at high speed for 30 seconds. Then return to pot, add chicken and proceed as before. As an appetizer: Looks lovely served individually in six wide, shallow soup bowls. Great starter for many kinds of meals, not just Thai. As a main course: By doubling the amount of chicken and keeping the same amount of liquid, you convert this soup practically into a stew. The six wide, shallow bowls still work well; place a half-cup of steamed white rice to one side in each bowl, then divide the soup/stew among the bowls. Serve a tomato salad on the side that has been dressed with lime juice, sugar, fish sauce, and topped with chopped peanuts and minced cilantro. If you can get the Thai ingredients: for the lime and lemon peel, substitute 3/8 cup Keffir lime leaves, 2 lemon grass stalks chopped, 3 thick slices dried galanga root.