Position a cooling rack onto a baking sheet and then lay eggplant slices on top; sprinkle with salt and let stand for about 30 minutes (alternately, place salted slices in a colander and place in a sink, where they will drain).
Meanwhile, in two shallow bowls, place bread crumbs in one and eggs in the other.
Preheat oven to 400°F. Coat baking sheet with oil.
Pat dry eggplant with paper towel and sprinkle both sides with flour.
Starting with one slice, dip completely into eggs, then press into bread crumbs to coat well. Place on prepared sheet and repeat with the remaining slices, nestling all the slices in a single layer on the sheet.
Drizzle with some oil or coat with cooking spray and bake for about 10 minutes. Flip slices over and sprinkle with Parmigiano. Bake for 8 to 10 more minutes (if you're using mozzarella, divide it among the eggplant slices and bake for another 5-8 minutes or until the cheese is melted!)
Serve warm with a dollop of warm tomato sauce and a spoonful of pesto.