In a shallow bowl, stir together the flour, cornmeal, cayenne pepper, salt, and black pepper. Place the eggs in a separate bowl.
Heat the oil in a large skillet over medium heat. A few at a time, dip the eggplant pieces into the egg, then into the flour mixture, then back into the egg, and back into the flour mixture. Fry the eggplant in the hot oil until golden brown. Drain on paper towels, and serve immediately.
