1. Place the flour, eggs, and bread crumbs in separate shallow bowls. Dip the eggplant slices in the flour, then in the eggs (letting any excess drip off), and finally in the bread crumbs, pressing gently to help them adhere.
2. Heat half the oil in a large nonstick skillet over medium-high heat. Cook half the eggplant until golden brown and crisp, 2 to 3 minutes per side; transfer to a paper towel–lined plate. Wipe out the skillet and repeat with the remaining oil and eggplant. Sprinkle the hot eggplant with ½ teaspoon salt.
3. Layer the eggplant, lettuce, tomatoes, basil, and goat cheese to form 4 "sandwiches," with the eggplant on the top and bottom of each stack.
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