Slice the eggplants into ½-inch-thick pieces and layer in a colander, salting the slices heavily as you go. Set aside for 30 minutes.
Combine the ricotta, eggs, Parmigiano-Reggiano, and chopped parsley. Season with salt and pepper.
Rinse the eggplant slices well and pat dry with paper towels. Heat 2 tablespoons olive oil in a large skillet over medium-low heat until it begins to smoke. Add a single layer of eggplant slices with no overlapping. Turn the slices quickly to coat both sides lightly with oil; reduce the heat slightly. Fry the eggplant until lightly browned on both sides. (Do not add more oil after the eggplant is in the skillet.) When the slices are browned, remove to paper towels to drain. Pour 2 tablespoons more oil into the skillet and cook another layer of eggplant. Repeat until all the eggplant pieces are done.
Spread ½ cup of the tomato sauce over the bottom of an oval gratin dish measuring 9 × 12 inches. Arrange a layer of eggplant slices over the sauce. Top each eggplant slice with a tablespoon of ricotta mixture and sprinkle about one third of the grated mozzarella over the layer. Repeat, arranging the next layer of eggplant slices to cover the gaps between the slices in the first layer. Add more ricotta mixture and mozzarella. Add a final layer of eggplant, cover it well with the remaining tomato sauce, and spoon the remaining ricotta mixture down the center of the dish. Sprinkle the remaining mozzarella over the exposed tomato sauce.
Set the dish on the center rack of the oven and bake until well browned and bubbling, 25 to 30 minutes. Let stand for 10 minutes before serving.