2 pounds eggplants, preferably 6 to 8 ounces each
1 tablespoons kosher salt, plus 3/4 teaspoon
1½ pounds plum tomatoes
1 pound young Pecorino , or low moisture mozzarella cut in 1/2-inch cubes
¼ cup fresh basil leaves, chopped
¼ teaspoon peperoncino, or to taste
½ cup extra-virgin olive oil
2 large onions, peeled, sliced in 1/2-inch rounds
6 garlic cloves, crushed and peeled
1 cup bread crumbs, dry
1½ tablespoons Italian parsley, chopped
1 tablespoon fresh thyme, chopped
½ teaspoon dried oregano
2 pinches of kosher salt
¼ cup pecorino, freshly grated
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, to press out the vegetable liquid. 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. Trim the stem end of the plum tomatoes, and cut them in half crosswise. Squeeze the halves over a strainer set in a bowl, scraping out and catching the seeds in the strainer and collecting the tomato juices in the bowl. Cut the tomatoes into 1/2-inch chunks, put them all in a bowl, and toss with the cubes of cheese, chopped basil, peperoncino, the remaining 3/4 teaspoon salt, and 4 tablespoons of the olive oil. Brush the bottom and sides of a baking dish with about 2 tablespoons of the olive oil. Lay the onion rounds in the bottom in one layer, with the garlic cloves scattered among them, and drizzle over them the remaining 2 tablespoons olive oil. Lay the eggplant slices over the onions in an even layer, overlapping a bit if necessary. Scatter the tomato and cheese chunks evenly on top of the eggplant, and drizzle the collected tomato juices all over. Put the bread crumbs and all the dry seasonings in a bowl, toss to blend, then drizzle over them the 2 tablespoons olive oil, and toss well, until the crumbs are evenly moistened. Sprinkle them evenly over the top of the eggplant, tomato, and cheese layer.
Tent the baking dish with a sheet of heavy aluminum foil, arching it so it doesn't touch the crumb topping, and 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 30 minutes or so, until the eggplant is lightly caramelized on the edges but tender and easily pierced with the tip of a knife, and the bread-crumb topping is browned and crisp.
Let the eggplant rest for 20 minutes before serving (it will absorb the sauce, and the cheese will settle a bit, too). Serve as you would lasagna, cutting portions with a knife and lifting them out with a wide spatula into warm shallow bowls. If you like, sprinkle shredded fresh basil over each portion.