Preheat oven to 450F. Line a baking sheet with aluminum foil or parchment paper, and brush with olive oil. Slice the eggplants lengthwise down the center, being careful not to cut all the way through the skin on the other side. Set on the baking sheet and bake 20 minutes, until the outer skin shrivels up a bit. Remove from oven. Transfer, cut side down, to a colander set over the sink. Let drain for 30 minutes.

Heat 2 tablespoons of the olive oil over medium heat in a large skillet with a lid. Add the onions and cook, stirring often, until tender, 5-8 minutes. Stir in the garlic, and cook, stirring, for 30 seconds to 1 minute, until fragrant. Remove from heat and transfer to a large mixing bowl. Add the tomoatoes, parsely, dill, basil, salt to taste, 1 teaspoon of the sugar and 1 tablespoon of the olive oil to the bowl and stir to combine. In a separate small bowl, whisk together the remaining olive oil, sugar, water and lemon juice (if using).

Set the drained eggplants in the skillet, cut side up. Season the eggplant with salt, then scoop the onion and tomato mixture into the eggplants. Drizzle the lemon-water-olive oil mixture around the eggplants. Cover the skillet and cook over low heat for 1-1 1/2 hours, checking the pan for liquid and basting the eggplants from time to time. Add a little water if needed. When the eggplant is done, they should be very tender and almost flattened, and the liquid will be caramelized. Spoon the juice over the eggplants and let cool to room temperature. Serve.