Trim off the stem end from the eggplant and cut it crosswise into slices about 1/2 inch thick. If you have some extra time, put the eggplant in a colander in the sink, sprinkle with 1 tablespoon salt, and toss to coat the slices on both sides. Let the eggplant rest for at least 20 minutes and up to 1 hour.
Heat the oven to 200°F. Beat the eggs with some salt and pepper in a shallow bowl or pie plate. Put the flour and bread crumbs on separate plates, with the eggs in between them. Have a baking sheet and several rectangles of wax or parchment paper ready.
If you salted the eggplant, rinse and dry it well with paper towels. Coat a slice in the flour, dip in the egg, then coat in the bread crumbs. You want a thin, even layer of each coating; shake off any excess. Spread the eggplant on the baking sheet in a single layer, top with wax or parchment paper, and repeat with the remaining slices. Transfer the pan to chill in the refrigerator for at least 10 minutes and up to 3 hours.
Put a large skillet over medium heat and pour in enough oil to come about 1/2 inch up the sides. While the oil heats, line a plate with paper towels. The oil is ready when a pinch of flour sizzles immediately.
Put a few of the eggplant slices in the hot oil without crowding the pan. When the bottoms turn brown, after 2 to 3 minutes, turn them over and cook the other side for 2 to 3 minutes, adjusting the heat to keep the oil sputtering without smoking or burning the eggplant. As each piece is done, put it on the paper towels to drain, turning them if the tops need blotting too. Transfer them to an ovenproof platter and keep them warm in the oven while you finish cooking the remaining eggplant. In between batches, add more oil to maintain a depth of 1/2 inch and let it heat up a bit.
When all the pieces are cooked, garnish with the parsley and serve with the lemon wedges.