Step 1Preheat the broiler or prepare the grill with medium-hot coals.
Step 2Prick the eggplants deeply 4 or 5 times with a fork. Broil or grill the eggplants, turning them a few times, until they are very tender, about 20 minutes. When they are done, they should look collapsed and their skin will be black. Check for tenderness by pressing on the neck end; or you can pierce it with a fork -- it should go in very easily.
Step 3While the eggplants are cooking, prepare the tomatoes: Bring a large saucepan of water to a boil over high heat. Core the tomatoes, and slit the skin at the bottom of each tomato in an X. Put the tomatoes into the boiling water to cover and boil just until the tomato skin starts to pull away from the X, about 30 seconds. Transfer the tomatoes to a bowl of ice water to stop the cooking. Remove them from the cold water promptly and pull off the peel. Halve the tomatoes and squeeze out their juice and seeds, then chop them. You should have 1 1/4 cups. Set aside.
Step 4Remove the eggplants from the oven or grill, and cut off the eggplant caps. Halve the eggplants lengthwise. Scoop and scrape out the eggplant pulp from inside the skin. Chop the eggplant pulp fine; you should have about 1 1/2 cups pulp.
Step 5Heat the oil in a large heavy skillet. Add the onion and saute over medium heat, stirring occasionally, until it has softened and is beginning to brown, about 10 minutes. Add the garlic and saute for 5 to 10 seconds, stirring. Stir in the turmeric, followed by the tomatoes, salt and pepper. Cook over medium heat, stirring often, until the mixture is thick, about 7 minutes.
Step 6Add the chopped eggplant and cook over medium-low heat, stirring often, until the mixture is thick and the flavors are blended, about 5 minutes. If you like, stir in the tomato paste and heat, stirring, until blended. Taste and adjust seasoning; season generously.
Step 7Serve the eggplant warm, at room temperature or cool, spooned into a shallow bowl and sprinkled with parsley. Garnish with hard-boiled egg quarters and black olives.