Prepare the Eggplant: Place the sliced eggplant in a colander and sprinkle with the kosher salt. Allow to sit for 15 minutes, then rinse and place on a double layer of paper towels and pat dry.
Place the lightly beaten eggs in one shallow bowl, and the bread crumbs in another (I like using pie plates for this!). Dip the slices of eggplant into the egg, allowing any excess to drip off, then coat in the bread crumbs. Place the breaded eggplant on a baking sheet while you prepare the rest.
Pour the olive oil and vegetable oil in a large, deep skillet (I use a 12-inch cast iron skillet) and heat over medium-high heat. Add a pinch of bread crumbs to see if the oil is ready - if they begin bubbling and sizzling the oil is ready.
Add the eggplant slices a few at a time (do not overcrowd the pan, four is about the max amount per batch) and cook until golden brown, about 2 minutes per side. Remove to a paper towel-lined baking sheet to drain.
Assemble the Eggplant Parmesan: Preheat oven to 350 degrees F.
Spread ¾ cup of the pasta sauce over the bottom of a 9x13-inch baking dish. Cover the sauce with slices of eggplant (the pieces can overlap), then spread 1 cup of sauce over the eggplant slices, sprinkle with 2 cups of the shredded mozzarella cheese, and 1/3 cup of the Parmigiano-Reggiano cheese. Repeat another layer of eggplant, 1 cup of sauce, 2 cups shredded mozzarella and 1/3 cup Parmigiano-Reggiano cheese. For the last layer, add the sliced eggplant, 1 cup of sauce, remaining 1 cup shredded mozzarella, and 1/3 cup Parmigiano-Reggiano cheese.
Cover the pan with foil and bake for 30 minutes. Remove the foil, increase the oven temperature to 375 degrees F, and continue to bake for another 10 to 15 minutes, or until the cheese on top is melted, browned and bubbling. Allow to rest for 10 minutes before serving.