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).