Trim the stem and bottom ends of each eggplant and, with a vegetable peeler, shave off ribbons of skin lengthwise, creating a zebra-striped pattern all around. Slice the eggplants crosswise into 1/2-inch thick rounds. Lightly salt the slices on both sides, using about a tablespoon in all, and layer them in a colander, set in a bowl or the sink. Invert a dinner plate over the piled slices, and weight it with cans or other heavy objects. Let the rounds drain for 30 minutes to an hour, then rinse them, and pat dry with paper towels.
Meanwhile, arrange a rack in the middle of the oven, and heat to 400 degrees F.
Put the tomato sauce in a bowl, and stir in the olives, sliced pepperoncini capers, and basil. Spread about 1/2 cup of the sauce over the bottom of a baking dish (I used a 9 X 9 dish, but the extra thick layers required an extra 20 minutes of baking time). Lay in half the eggplant slices in a single layer, overlapping if necessary. Drop spoonfuls of tomato sauce, another 1/2 cup total, on top of the eggplant layer, and sprinkle over it half the grated cheese. Lay in the remaining eggplant rounds in an even layer, spoon and spread the remaining sauce over them, and scatter the rest of the cheese all over the top.
Tent the baking dish with a sheet of heavy aluminum foil, arching it so it doesn't touch the cheese topping, pressing it against the sides. Set the dish on a baking sheet, place in the oven, and bake for 45 minutes, until the juices are bubbling actively. Remove the foil, and continue baking for another 15 minutes or so, until the cheese topping is golden brown and eggplant is caramelized on the edges yet tender and easily pierced with the tip of knife.
Let the eggplant rest for 20 minutes (to absorb the sauce). Cut portions with a knife, and lift them out with a wide spatula into warm shallow bowls.