2 cups homemade breadcrumbs
6 large eggs
3/4 cup plus 2 tablespoons finely grated Parmigiano-Reggiano
1 cup all-purpose flour
2 to 3 cups olive oil, for frying, plus more for greasing baking dish
3 cups marinara sauce
12 ounces fresh mozzarella cheese, cut into about 12 to 14 1/4-inch thick slices
Cut the eggplant crosswise into 1/4-inch thick rounds; you will have about 20 rounds. Put the slices in a colander, salt liberally, toss and let rest in the sink for 1 hour to draw out the bitterness and excess moisture. Rinse briefly under cold running water, turning to expose all of the eggplant to the water, then drain, remove the slices from the colander, and pat dry with paper towels.
Position a rack in the center of the oven. Preheat to 375F.
Arrange a wire rack over a baking sheet. Line a large plate or platter with paper towels. Put the breadcrumbs in a wide shallow bowl. Put the eggs in another bowl with 1/2 cup of Parmigiano-Reggiano and whisk. Put the flour in the third bowl.
Working with 1 eggplant slice at a time, dredge the slices in the flour, then the eggs, then breadcrumbs. As they are prepared, gather them in a single layer on the wire rack.
Heat a wide, deep, heavy skillet over medium-high heat. Pour in oil to a depth of 1/4 inch, and heat until a few breadcrumbs added to the oil sizzle and brown.
Carefully add the eggplant to the oil in batches, without crowding, and fry until golden brown on both sides, 2- 3 minutes per side, turning the slices with tongs when the first side is browned. As they are done, use the tongs to transfer them to the paper towel-lined plate and season immediately with salt. Repeat until all the eggplant slices are fried, drained, and seasoned.
When the slices are cool enough to handle, lightly oil a shallow glass or ceramic 9x13 x2 baking dish. Ladle a few tablespoons of the marinara sauce onto the dish and use the back of the ladle to spread it and coat the bottom of the dish.
Arrange the eggplant slices in overlapping fashion in baking dish. Scatter 2 tablespoons of the Parmigiano-Reggiano over them and ladle more sauce over the cheese. Cover with a layer of mozzarella. Repeat with another layer of eggplant, 2 more tablespoons of Parmigiano-Reggiano, sauce, and mozzarella, and finish with the final 2 tablespoons of Parmigiano-Reggiano. (Although best baked right away, the eggplant Parmesan may be covered with plastic wrap and refrigerated overnight. Remove the plastic and let come to room temperature before baking.)
Bake until the cheese is hot and bubbling and the dish is hot throughout, 20 to 30 minutes. Let rest 10 minutes before slicing and serving.