Peel the eggplants and slice into ¼-inch-thick rounds. Arrange in a single layer on paper towel-lined baking sheets and lightly sprinkle with kosher salt. Let sit for 30–45 minutes to draw out excess moisture. Blot thoroughly with paper towels.
Place the flour in a shallow bowl. In a second shallow bowl, whisk the eggs.
Pour enough olive oil into a large skillet to reach about ¼ inch up the sides. Heat over medium to medium-high heat. Working in batches, lightly dredge each eggplant slice in flour, shaking off the excess, then dip into the beaten egg, allowing the excess to drip off.
Fry the eggplant for 1–2 minutes per side, until lightly golden and tender. Adjust the heat as needed to prevent the coating from browning too quickly. Transfer to a paper towel-lined baking sheet to drain. Add more olive oil to the skillet between batches as needed.
Preheat the oven to 350°F. Spread a thin layer of marinara sauce over the bottom of a 9 x 13-inch baking dish.
Arrange a layer of fried eggplant over the sauce, slightly overlapping the slices. Spoon a thin layer of marinara over the eggplant and sprinkle with Parmesan. Add a layer of mozzarella every other layer. Continue layering the eggplant, marinara, Parmesan, and mozzarella (every other layer) until all of the eggplant has been used.
Finish the final layer with marinara sauce, mozzarella, and Parmesan. Scatter the butter evenly over the top.
Bake uncovered for 35–40 minutes, until the sauce is bubbling and the cheese is melted and golden brown.
Let the Eggplant Parmesan rest for 10–15 minutes before slicing and serving.