Prep the aromatics. Peel the outer layers of the lemongrass and bruise the stalks with the flat of a knife. Cut into 2-inch pieces.
Slice galangal into thin coins. Tear kaffir lime leaves to release their oils.
Warm the aromatics. In a large pot, heat the oil over medium. Add lemongrass, galangal, shallot, and kaffir lime leaves.
Stir for 2–3 minutes until fragrant but not browned.
Add broth and simmer. Pour in the chicken broth. Bring to a gentle boil, then lower to a simmer. Let it bubble for 10 minutes to infuse the flavors.
Add coconut milk. Stir in the coconut milk and return to a gentle simmer.
Avoid a rolling boil to keep the coconut milk smooth.
Poach the chicken. Add the thin-sliced chicken and mushrooms. Simmer gently 6–8 minutes until the chicken is just cooked through. Skim any foam if needed.
Season the soup. Add fish sauce 1 tablespoon at a time, tasting as you go.
Stir in a small amount of sugar if you want a touch of roundness. Add sliced chiles to your heat preference.
Balance with lime. Turn off the heat. Stir in lime juice, starting with 2 tablespoons and adding more until the flavor is bright and balanced.
You’re looking for a gentle tug of acidity, not sharpness.
Finish and serve. Remove the lemongrass, galangal slices, and lime leaves if you prefer. Ladle into bowls, then top with cilantro and green onions. Serve hot with extra lime wedges and fish sauce on the side.