Preheat oven to 450°F. Coat an 8-inch-square glass baking dish with cooking spray.
Whisk egg in a shallow dish. Place breadcrumbs in another shallow dish. Dip eggplant slices in the egg and then dredge in the breadcrumbs to coat thoroughly; transfer to a plate.
Heat 2 teaspoons oil in a large nonstick skillet over medium-high heat. Add half the eggplant slices and cook until browned on the first side, about 2 minutes. Turn the slices, add 1 teaspoon oil and reduce heat to medium; cook until browned on the second side, about 2 minutes. Transfer the cooked eggplant to the prepared baking dish. Repeat with the remaining oil and eggplant; transfer the cooked slices back to the plate.
Wipe out the pan; add tomato sauce and salt, and bring to a simmer over medium-high heat, stirring constantly. Remove from the heat and stir in basil. Pour half the sauce over the eggplant in the baking dish. Sprinkle with half the cheese. Layer on the remaining eggplant. Top with the remaining sauce.
Cover the baking dish with foil. Bake until bubbling, about 10 minutes. Remove the foil, sprinkle with the remaining cheese and continue baking until the cheese is melted, 3 to 5 minutes.