2 (1-pound) eggplants
6 tablespoons olive oil, divided, plus additional for drizzling
3/4 cup plain dry bread crumbs
1/2 cup grated Parmigiano-Reggiano
1/2 cup finely chopped flat-leaf parsley
2 garlic cloves, minced, divided
5 eggs, lightly beaten
1/2 cup water
1/4 teaspoon hot red-pepper flakes
1/4 pound arugula, coarse stems discarded, coarsely chopped
1 cup packed basil leaves, coarsely chopped
1/2 pound cold fresh mozzarella, cut into 4 (1/2-inch-thick) slices
Preheat oven to 400°F.Cut 12 (1/3-inch-thick) rounds from widest portion of eggplants. (I peel the skins - not shown above). Brush both sides with 2 tablespoons oil and season with 1/2 teaspoon salt (total). Bake on an oiled baking sheet, turning once, until golden and tender, 20 to 30 minutes. Transfer to a plate and keep warm, covered. Leave oven on.Make egg patties and sauté arugula:
Stir together bread crumbs, parmesan, parsley, half of garlic, and 1/4 teaspoon each of salt and pepper, then stir in eggs and water.
Heat 3 tablespoons oil in a 12-inch heavy skillet over medium heat until it shimmers. Drop about 2 tablespoons of egg mixture into skillet and cook, turning once, until patties are golden brown and puffed, about 5 minutes. Transfer to paper towels to drain.
Add remaining tablespoon oil to skillet and cook remaining garlic with red-pepper flakes, stirring, until garlic is golden, about 30 seconds. Add arugula and basil and stir until just wilted, then stir in 1/8 teaspoon salt.
Arrange egg patties on a baking sheet. Top each with tomato sauce, 1 slice mozzarella, 1 eggplant slice, 2 more tablespoons tomato sauce, another eggplant slice, arugula mixture, and remaining eggplant. Bake until cheese melts, 5 to 10 minutes. Drizzle with additional oil and serve remaining sauce on the side.