To prepare the eggplant: Pour 3 quarts of water into a very large bowl or large pan. Add the salt and stir well.
Trim the eggplants, slicing off the very top and bottom, and then peel them. Cut them, crosswise, into about 5 slices each. Each slice should be about 1 1/4 to 1 1/2 inches thick. Put the slices into the salty water as you cut them. The slices will float. Upturn a plate on top of the slices and balance a weight (such as a clean glass jar filled with water) on top of the plate. Set aside for 3 to 10 hours.
Remove the slices from the water and pat them dry.
Pour about enough oil into a large pan to come to a depth of 3/4 inches, and set over medium heat. Allow time for the oil to get very hot, then put in as many eggplant slices as the pan will hold in a single layer and fry for 3 to 4 minutes on each side, or until reddish-gold. Remove to a large plate or tray lined with a double thickness of paper towels. Fry all the eggplant slices this way, adding more oil, if needed, and using as many plates or trays as you see fit. I like to change the paper towels at least once. I also pat the tops of the slices with more paper towel. The eggplants may now be set aside for 3 to 4 hours, if desired.
To make the Tomato-chickpea sauce: Either use 3 tablespoons from the frying oil or, if you prefer, fresh oil, and pour it into a large, lidded pan. Set the pan over medium-high heat, and, when the oil is very hot, put in the chiles and mustard seeds. As soon as the mustard seeds begin to pop, a matter of seconds, add the garlic. Stir quickly once, then put in the tomatoes, curry leaves, chickpeas, salt, cumin, coriander, and turmeric. Stir and bring to a simmer. Cover, reduce the heat to low, and simmer gently for 20 minutes. Remove the chiles before serving.
To make the yogurt sauce: Put the yogurt in a small bowl and beat lightly with a fork or small whisk until smooth. Stir in the salt and cumin. Cover and refrigerate, if not using within 2 hours.
To make the tamarind chutney: Combine the tamarind paste, sugar, salt, and cumin in a small cup. Mix well. Cover and refrigerate, if not using within 2 hours.
To prepare the dish: When ready to serve, preheat the oven to 325 degrees F. Spread out the eggplant slices in a single layer on a baking tray and dust very lightly with salt and black pepper. Place in the oven and heat through, about 15 minutes. If oil accumulates at the bottom of the tray, pour it out.
Heat the tomato-chickpea sauce.
Arrange the eggplant slices on a very large platter in a single layer. Pour a small ladleful of the tomato-chickpea sauce on top of each slice. Center 2 tablespoons of the yogurt sauce on top of the tomato-chickpea sauce and then put a generous dot of the tamarind chutney on top of the yogurt. Garnish with mint leaves.
Recipe courtesy of Madhur Jaffrey, From Curries to Kebabs: Recipes from the Indian Spice Kitchen, Clarkson Potter, 2003