Position rack in middle of the oven and heat broiler to low. Place eggplant slices in a colander set in the sink or over a bowl, sprinkle generously with kosher salt, toss to combine, and set aside to drain while you prepare the other ingredients.
Place flour in a wide, shallow dish and season generously with kosher salt and freshly ground black pepper. Combine eggs and milk in another shallow dish and set aside; mix panko and cheese together in a third shallow dish.
Remove eggplant slices from colander and pat dry with paper towels. Bread eggplant by coating a few slices in flour mixture. Shake off excess flour, dip into egg mixture, and press into panko mixture; be sure to coat the slices thoroughly at each step. Set breaded eggplant on a baking sheet and repeat with remaining eggplant.
Line another baking sheet with paper towels and set aside. Fill a large, straight-sided skillet or frying pan with 1 inch oil. Warm over medium-high heat until oil reaches 350°F on a deep-fat thermometer (oil will be shimmering but not smoking). Add about 1/3 of eggplant slices and fry on one side until golden brown, about 2 minutes. Carefully flip over and fry other side for another 2 minutes, or until golden brown. Remove to paper towel–lined baking sheet and repeat with remaining eggplant.
Pour warmed tomato sauce over the bottom of an 8-by-8-inch baking dish. Shingle one eggplant slice, one basil leaf, and one mozzarella slice over the tomato sauce, repeating until all ingredients are arranged in the baking dish.
Place dish in the oven and broil until cheese is melted, bubbly, and speckled with gold, about 5 to 7 minutes. Serve immediately with extra tomato sauce on the side.