In a medium-sized pot or Dutch oven, combine the chicken broth, smashed lemongrass, sliced galangal, and torn kaffir lime leaves. Bring the mixture to a gentle boil over medium-high heat, allowing the aromatics to infuse the broth for about 5 minutes until very fragrant.

Stir in the coconut milk, fish sauce, and sugar, then lower the heat to medium. Simmer gently for another 5 minutes, ensuring the liquid doesn't come to a rolling boil so the rich coconut milk remains smooth and does not separate.

Add the sliced chicken and quartered mushrooms to the pot, stirring gently to combine. Continue to simmer for 8 to 10 minutes, or until the chicken is completely cooked through and no longer pink inside, and the mushrooms are tender.

Remove the pot from the heat and stir in the fresh lime juice and bruised chilies. Tasting and adjusting the seasoning off the heat ensures the lime juice retains its bright, fresh acidity without turning bitter.

Ladle the hot soup into serving bowls, taking care to avoid eating the tough pieces of lemongrass and galangal. Garnish generously with fresh cilantro and serve immediately, alongside a bowl of steamed jasmine rice if desired.