Pour about half the coconut milk into a medium saucepan over medium heat. Add the galangal, lemongrass, and kaffir lime leaves.
Warm gently — bring just to the edge of a simmer, not a rolling boil. Let the aromatics infuse 8 minutes, stirring occasionally and scraping the bottom so the coconut milk does not scorch.
Add the chicken pieces. Keep the heat at a gentle simmer and cook 5 minutes, until the chicken is cooked through to 165°F internal with no pink.
Add the mushrooms and chiles; simmer 2 minutes more until the mushrooms are tender.
Stir in the remaining coconut milk, the fish sauce, and the sugar. Warm through gently — do not let it come to a hard boil.
Take the pot off the heat. Stir in the lime juice; taste and adjust fish sauce, sugar, and lime to balance. Serve in bowls topped with cilantro, with jasmine rice alongside. Leave the woody lemongrass, galangal slices, and lime leaves in the pot or pushed aside in the bowl — they are flavoring, not food.