Trim away and discard the root end and the top 3 inches of each stalk of lemongrass, along with any brittle leaves. Pound each stalk lightly with the spine of a cleaver or an unopened can. Cut each stalk crosswise into 2-inch lengths and set aside.
In a large serving bowl, combine the lime juice, fish sauce, scallions, and half of the wild lime leaves. Set the bowl by the stove, along with small dishes containing the galangal, lemongrass, and remaining lime leaves; the chiles (if using), the chopped cilantro; and the sliced chicken and mushrooms.
In a medium saucepan, combine the coconut milk and broth. Bring to a gentle boil over medium-high heat. Stir in the galangal, lemongrass, and lime leaves. Add the chicken and mushrooms. Return to a gentle boil, reduce the heat, and simmer for 10 minutes to infuse the flavors and cook the chicken.
Remove the pan from the heat, pour the hot soup over the seasonings in the serving bowl, and stir well. Sprinkle with the chopped cilantro and serve hot.