1. In a wok or large pot, combine 1 cup of the coconut milk with the galangal, lemongrass, and lime leaves and bring to a boil. Add the chicken, fish sauce, and sugar, reduce the heat to medium, and simmer until the chicken is white and firm, about 4 minutes. Add the remaining coconut milk and heat to just below boiling, about 3 minutes.
2. Divide the lime juice and curry paste into individual serving bowls and ladle the soup over them. Garnish each bowl with the cilantro and crushed chile peppers. Serve immediately.