Preheat the oven to 350°F. Lightly oil a rimmed baking sheet.
Leaving the peel on the eggplant, slice them into ½-inch slices. Place the slices in a colander, sprinkling with kosher salt as you layer in the slices. You will use about 1 teaspoon of kosher salt in total. Place the colander in the sink and let the eggplant sit for 30 minutes; the salt is pulling out the excess moisture from the eggplant, which allows it to cook up more firmly and crisp up better, and also removes any bitterness.
Place the flour on a plate, and place the eggs in a shallow bowl. Mix the breadcrumbs with the oregano, 1 teaspoon kosher salt, and the pepper on a separate plate. Pat the eggplant dry with a clean dish towel or paper towels (do not rinse the eggplant!).
Place a couple of wire racks, or baking sheets nearby. Place each slice of eggplant on the plate with the flour, then flip it over, but don’t press down on the eggplant; you just want a light coating of flour on both sides of the eggplant. Dip each slice into the beaten eggs, then allow any excess egg to drip back into the bowl. Place the eggplant slices on the plate with the breadcrumbs, turn it to coat both sides. Place the coated eggplant on the wire rack.
Heat 3 tablespoons the oil in a very large skillet over medium heat. Sauté the breaded eggplant slices in batches in a single layer for about 3 to 4 minutes on each side until the eggplant is browned on the outside, and a bit tender on the inside. Transfer the fried eggplant to paper towels to drain as they are finished, and continue frying the eggplant, adding more oil to the pan as you go.
Drizzle a thin layer of tomato sauce into the bottom of the oiled baking sheet. Arrange the eggplant over the sauce, overlapping the slices slightly so that they all fit in as close to a single layer as possible, but don’t worry about the overlapping. Place the thin slices of the mozzarella over the eggplant, and sprinkle them with the Parmesan. Bake for about 20 to 25 minutes until the cheese is melted and a touch browned.
Remove from the oven and spoon over the warmed sauce. Serve hot.