Partially peel the eggplant lengthwise, leaving 1/2-inch stripes of skin. Remove the stem. For large eggplant, quarter it lengthwise first. Slice the eggplant just thicker than 1/4 inch. Sprinkle generously with salt and let stand in a colander for about 1 hour. Rinse the eggplant thoroughly and firmly squeeze each slice dry, patting with paper towels. In a large skillet over high heat, warm about 1/2 cup olive oil until very hot. Fry the eggplant slices a single layer at a time until rich brown on both sides, 1 to 2 minutes per side; use more oil if needed. Drain on paper towels.

Discard the oil from the pan and wipe it clean. Warm 1 tablespoon olive oil over medium heat. Add the tomatoes and sprinkle with a little salt; simmer, mashing the tomatoes with a fork, until they form a sauce and are no longer liquidy, about 15 minutes. Stir in the garlic and vinegar; remove from the heat.

Fan the eggplant slices on a platter and coat with the tomato sauce. Let cool and then refrigerate for at least a day. Serve cold or at room temperature.