Arrange eggplant in a large colander and season generously with salt. Let stand 30 minutes, then rinse and pat dry.
Divide flour, egg mixture, and breadcrumbs among 3 shallow bowls. Dip each eggplant slice, turning to coat and shaking off excess, in the flour, then egg mixture, then breadcrumbs.
Heat oil in a 12-inch skillet (oil should be about 1/2-inch deep) over medium heat, and, working in batches, fry eggplant until golden and crisp, about 4 minutes per side.
Transfer to a paper-towel-lined plate and season with salt and pepper.
Top each cutlet with arugula, Parmesan, and a squeeze of lemon.
To check whether the oil is ready, drop in a few breadcrumbs; they should sizzle immediately.