Heat vegetable oil in a large dutch oven over medium heat. Stir in onion and curry paste until fragrant and onions begin to soften, about 3 minutes. Add coconut milk and water; bring to a boil. Reduce heat and simmer, uncovered, until liquid begins to thicken, about 5 minutes.

Mix in sweet potato and cook until softened, about 6 minutes more. Add eggplant, Thai chile peppers, and kaffir lime leaves. Cook and stir until vegetables are very tender, about 10 minutes.

Stir in lime juice, fish sauce, brown sugar, and lime zest. Mix thoroughly to combine and heat through, about 5 minutes more. Sprinkle with cilantro and Thai basil.