Heat the peanut oil in a deep pot with lid, over medium heat. Add the garlic, lemongrass and turmeric to the pot and stir.
Add the tomatoes to the pot. Stir in 2 tablespoons water. Simmer until the tomatoes break down, about 2 minutes.
Stir in 1 tablespoon fish sauce and ½ teaspoon sugar.
Add the eggplant to the pot. Pour in 1 cup water.
Add the remaining fish sauce and sugar. Stir in the chili and season with salt, and pepper. Simmer until the eggplant is soft, about 5 to 10 minutes.
Garnish with fresh basil and sliced green onions.