In a large pot, bring the chicken broth to a boil.
Add the lemongrass, galangal, and kaffir lime leaves. Reduce the heat and simmer for about 10 minutes to infuse the flavors.
Add the thinly sliced chicken to the pot and cook until it is no longer pink, about 5-7 minutes.
Pour in the coconut milk and bring the mixture back to a simmer.
Add the sliced mushrooms, tomato wedges, and Thai bird's eye chilies. Cook for another 5 minutes until the vegetables are tender.
Stir in the fish sauce and sugar. Adjust the seasoning to taste, adding more fish sauce or sugar if needed.
Remove the pot from the heat and stir in the lime juice.
Ladle the Tom Kha Gai into serving bowls.
Garnish with fresh cilantro and serve with lime wedges on the side.
Enjoy your meal!