Preheat the oven to 450°. Set the eggplants on a large rimmed baking sheet and pierce all over. Roast the eggplants for 45 minutes, until very soft.
Meanwhile, peel the tomatoes with a vegetable peeler. Working over a coarse sieve set over a bowl, cut the flesh from the tomatoes and transfer to a shallow bowl. Allow the seeds, core and juices to fall into the sieve. Crush the tomato cores and press the juice through the sieve. Add the juice to the bowl with the tomato flesh; discard the seeds and cores. Season the tomato with 1/2 teaspoon of salt and let stand until more juices are released, about 25 minutes.
Roast the peppers directly over a gas flame or under the broiler, turning, until charred all over. Transfer the peppers to a bowl and let cool. Peel, core, seed and coarsely chop the peppers.
Cut the tomato sections into small dice. Return them to the bowl and stir in the chopped peppers, garlic, vinegar, sugar and the 1/4 cup plus 1 tablespoon of olive oil. Season with pepper.
Peel the eggplants, leaving the stem attached. Cut down the center of each eggplant and discard any large seed sacs; lightly drizzle with olive oil. Season well with salt. Spoon the tomato dressing over the eggplants. Sprinkle with the basil and serve.