Cut the chicken breast fillet into bite-sized pieces and clean the shiitake mushrooms before slicing. Wash the bok choy, cut into large chunks and slice the red chili pepper into thin rings. Crush the lemongrass with the flat side of a knife and slice the galangal into thick pieces so the flavors can release well.
Pour the chicken stock together with the coconut milk into a soup pot and bring to a boil over medium-high heat. Add the lemongrass, galangal and lime leaves to the hot liquid. Add the chicken breast pieces to the pot and let simmer gently over low heat for ten minutes until the meat is completely cooked.
Add the prepared shiitake mushrooms, bok choy and red chili pepper to the simmering broth. Let the soup simmer for another three minutes so the mushrooms become tender and the bok choy remains al dente. Remove from heat and stir in the fish sauce and lime juice to season the soup to taste.
Carefully remove the lemongrass, galangal pieces and lime leaves from the pot with a spoon and discard, as they are too tough to eat. Ladle the hot soup directly from the pot into large bowls and serve immediately.