Let the chicken broth boil together with the coconut milk. Simmer galangal, lemongrass (squeeze a little so that the taste can develop nicely), shallots and kaffir lime leaves for about 5 minutes. Then add the chicken breast and chilli pepper and simmer until the meat is through. Now add the mushrooms and season with the fish sauce, lime juice and sugar.
Vegetarian option: replace the chicken with more mushrooms or tofu.