Trim lemongrass as directed; pound lightly and cut into 2-inch lengths.
In a large serving bowl, combine lime juice, fish sauce, scallions, and half of the lime leaves. Set by the stove along with small dishes containing galangal, lemongrass, remaining lime leaves, chiles (if using), chopped cilantro, and the sliced chicken and mushrooms.
In a medium saucepan, bring coconut milk and broth to a gentle boil. Stir in galangal, lemongrass, and lime leaves. Add chicken and mushrooms; return to a gentle boil, reduce heat, and simmer 10 minutes to infuse flavors and cook chicken.
Remove from heat and pour the hot soup over the seasonings in the serving bowl; stir well. Sprinkle with chopped cilantro and serve hot. Pass around chiles for those who want them.