Slice the chicken breasts thinly. Bruise the lemongrass stalks with the back of a knife and chop them into 2-inch pieces. Tear the kaffir lime leaves. Thinly slice the red chilies. Slice the onion, mince the garlic, and prepare the remaining ingredients.
In a large pot, bring the chicken broth to a gentle simmer over medium heat.
Add the lemongrass, galangal slices, torn kaffir lime leaves, sliced chilies, sliced onion, and minced garlic to the pot. Let the broth simmer for 5-7 minutes.
Add the thinly sliced chicken breast to the pot and simmer for an additional 5-7 minutes, or until the chicken is cooked through.
Pour in the coconut milk and stir well to combine with the broth and chicken. Let the soup simmer gently for another 5 minutes.
Stir in the fish sauce, lime juice, and palm sugar. Taste the soup and adjust the seasoning, adding more fish sauce for saltiness, lime juice.
Add the sliced mushrooms and tomato wedges to the pot. Simmer for 2-3 minutes.
Stir in the chopped cilantro and Thai basil leaves.
Ladle the Tom Kha Gai into serving bowls. Garnish with additional fresh herbs and red chili slices if desired.