Heat 1/2 cup coconut milk (reserve the rest for later) in a skillet over medium heat. Stir until it comes to a boil, about 3–5 minutes. Add the green curry paste and cook, stirring, until the mixture thickens and becomes fragrant. Transfer everything to a large pot.

Place the large pot over medium heat. Add the chicken and stir for about 2 minutes. Add the fish sauce and sugar, and stir for another minute. Add the chopped Thai eggplants, the remaining coconut milk, and chicken broth. Simmer until the chicken is cooked through and the eggplants are tender.

Add the kaffir lime leaves and Thai basil leaves. Bring to a boil, then turn off the heat. Serve hot with steamed jasmine rice and chili fish sauce on the side.