Heat coconut oil in a 5-quart Dutch oven. Saute sliced mushrooms, scallions, and garlic.
Meanwhile, chop the lemongrass and Galangal root, and crush or chop the Kaffir lime leaves.
Remove mushrooms, scallions, and garlic from the pot and set aside. Add lemongrass, Galangal, and lime leaves. Saute until fragrant (1 to 2 minutes). Add curry paste and cook one more minute.
Add 4 cups of water and the chicken. Bring to a boil, then simmer, covered, for 30 minutes.
Remove chicken from the pot, strain the broth. Discard lemongrass, Galangal, and lime leaves. Pour broth back into the pot, along with the mushrooms, scallions, and garlic. Shred the chicken and add to the pot. Discard the skin and bones.
Add coconut milk, lime juice, fish sauce, brown sugar, and salt. Simmer for 15 more minutes. Serve with chopped cilantro and sliced pepper (if desired).
Notes
Galangal root can be very tough and hard to slice, especially if it's old. The younger the root, the more tender it will be. If you can't slice it, try grating or pounding it.
You can make a vegetarian Tom Kha by adding tofu instead of chicken.
Sometimes Tom Kha is served with chili oil. The strong sesame flavor of some chili oils overwhelms the other more subtle flavors, so I've decided to leave it out. If you do decide to try it, use it sparingly.
Tom Kha is only slightly spicy from the Galangal root. To add more heat, I recommend serving it with sliced chili peppers on the side.
To make the soup even more filling, serve Tom Kha Gai over cooked ramen noodles.