Slice eggplant lengthwise into sections about 1/4 inch thick and remove the skin. Place the eggplant slices in a dish with sides, sprinkle with salt, and let sit 15 minutes to draw out some of the moisture.
In a bowl, combine the eggs, parsley, milk, garlic, salt, and pepper. Pour over the eggplant slices and place in the refrigerator for 30 minutes.
Preheat the oven to 400 degrees F (200 degrees C).
Cover a baking sheet with a thin layer of oil and place in the oven to heat.
Place the breadcrumbs in a wide dish, and dip each slice of eggplant in the breadcrumbs to coat.
Place the eggplant on the hot tray and bake about 5 minutes, until the bottom is browned.
Turn slices over, sprinkle with Parmesan cheese, and bake for another 5 minutes until the cheese has melted and the bottom is browned.
Distribute the eggplant slices among the 4 sandwiches, top with lettuce and tomato, and serve.