Heat 2 tablespoons of oil in a wide pot over medium heat. Add the ginger, onion, garlic, and green chilies. Saute until they become limp and aromatic, which usually takes about 2-3 minutes.
Add the chicken, fish sauce, ground pepper, and lemongrass to the pot. Stir well to coat the chicken with oil. Cover the pot with a lid to allow the juices from the chicken to come out. Cook for several minutes until the sides of the chicken are slightly browned.
Pour in 1 cup of water, cover the pot, and let it cook for 15 minutes.
Add the green papaya to the pot and mix it in. Then, pour in the coconut milk and add the chili labuyo. Cover the pot again and let it simmer over low heat for 10-15 minutes or until the papaya and chicken become tender. Add more water if needed.
Season with salt and pepper according to your taste. Finally, add the chili leaves and give it a stir. Cook for another 2 minutes, then turn off the heat.
Transfer the soup to individual serving bowls and enjoy it with rice.