Peel eggplant and cut in rounds. Arrange the eggplant slices in a colander or on paper towels. Sprinkle generously with kosher salt and set aside to let the bitter juices weep from the eggplant, about 1 hour.
In the meantime, beat together egg and milk, and prepare a dish of breadcrumbs. Transfer the eggplant to a colander in the sink, and rinse well under cold water. Blot with paper towels before dredging in egg-milk mixture and breading. Working in batches, fry in vegetable oil until nicely browned.
Once eggplant rounds are fried, lay one on the bed of greens. Top with one slice of mozzarella and one basil leaf. Repeat with each fried eggplant round. Sprinkle with pimentos and drizzle with balsamic vinegar, olive oil, salt, and pepper before serving.