In a large pot, combine the coconut milk and chicken broth. Bring to a gentle simmer over medium heat.
Add the smashed lemongrass, galangal slices, and torn kaffir lime leaves to the pot. Let it simmer for about 5 minutes to infuse the flavors.
Add the sliced chicken and mushrooms to the pot. Cook until the chicken is fully cooked, about 5-7 minutes.
Stir in the smashed bird's eye chilies, fish sauce, lime juice, and brown sugar. Adjust the seasoning to taste.
Remove from heat and discard the lemongrass and galangal slices.
Serve hot, garnished with fresh cilantro and lime wedges on the side.