Coat the outside of the eggplant with 1 teaspoon vegetable oil. Place on a baking sheet and bake in the oven until soft, 20 to 30 minutes. Allow to cool before coarsely mashing; set aside.
Heat 2 tablespoons vegetable oil in a large skillet. Fry the dried red chile peppers, cumin seeds, mustard seed, and asafoetida powder in the hot oil until the seeds start to splutter. Add the onions, green chile peppers, and curry leaves; cook and stir until the onions are soft and begin to brown, about 5 minutes. Stir the mashed eggplant, water, tamarind paste, jaggery, white sugar, turmeric, ground red pepper, and salt into the mixture; bring to a boil and cook until the mixture begins to thicken, 5 to 10 minutes.
