In a saucepan, combine the sliced galangal (or ginger), cilantro stems, torn kaffir lime leaves, halved lemongrass, and 4 cups of water; bring to a boil.
Stir in the fish sauce and lime juice, reduce the heat, and simmer gently for 10 minutes.
Strain out and discard the cilantro stems (roots) and any large aromatics if desired, leaving the infused broth in the pan.
Add the coconut milk and return to a gentle simmer for 2–3 minutes, then taste and adjust: add more coconut milk or water if too sour, or a dash more fish sauce if not salty enough.
Add the thinly sliced chicken to the simmering broth and cook just until the chicken is opaque, milky-looking, and cooked through, about 1–3 minutes depending on thickness.
Stir in the sliced red chile, then ladle the soup into bowls and sprinkle with the reserved cilantro leaves before serving.