Toss eggplant with 2 teaspoons salt in a colander set over a bowl, then let drain 30 minutes. While eggplant drains, cut an X in bottom of each tomato with a sharp paring knife and blanch tomatoes together in a 5-quart pot of boiling water 1 minute. Transfer tomatoes with a slotted spoon to a cutting board and, when cool enough to handle, peel off skin, beginning from scored end, with paring knife. Coarsely chop tomatoes, then coarsely puree in batches in a blender. Heat 3 tablespoons oil in a 5-quart heavy pot over moderately high heat until hot but not smoking, then add garlic and saute, stirring, until golden, about 30 seconds. Add tomato puree, basil, 1 teaspoon salt, 1/2 teaspoon pepper, and red pepper flakes and simmer, uncovered, stirring occasionally, until slightly thickened, 25 to 30 minutes. Put oven rack in middle position and preheat oven to 375°F. Stir together flour, remaining 1/4 teaspoon salt, and remaining 1/4 teaspoon pepper in a shallow bowl. Lightly beat eggs in a second shallow bowl, then stir together panko and 1/3 cup Parmigiano-Reggiano in a third shallow bowl. Working with 1 slice at a time, dredge eggplant in flour, shaking off excess, then dip in egg, letting excess drip off, and dredge in panko until evenly coated. Transfer eggplant to sheets of wax paper, arranging slices in 1 layer. Heat remaining 1 1/2 cups oil in a deep 12-inch nonstick skillet over moderately high heat until hot but not smoking, then fry eggplant 4 slices at a time, turning over once, until golden brown, 5 to 6 minutes per batch. Transfer with tongs to paper towels to drain. Spread 1 cup tomato sauce in bottom of a rectangular 3 1/2-quart (13- by 11- by 2-inch) baking dish. Arrange about one third of eggplant slices in 1 layer over sauce, overlapping slightly if necessary. Cover eggplant with about one third of remaining sauce (about 11/4 cups) and one third of mozzarella. Continue layering with remaining eggplant, sauce, and mozzarella. Sprinkle top with remaining 1/3 cup Parmigiano-Reggiano. Bake, uncovered, until cheese is melted and golden and sauce is bubbling, 35 to 40 minutes.