Heat oil in a heavy medium saucepan over medium heat until it shimmers, then cook onion and garlic, stirring occasionally, until softened, about 6 minutes.
Meanwhile, blend tomatoes with juice in a blender until almost smooth.
Add to onion mixture in saucepan with water, sugar, and 1/4 teaspoon salt and simmer, partially covered, stirring occasionally, until slightly thickened, about 10 minutes. Stir in basil and keep warm, covered.
Preheat oven to 450°F with rack in lowest position.
Cut 12 (1/3-inch-thick) rounds from widest portion of eggplants.
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.
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 4 rounded 1/3 cups 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 4 egg patties about 3 inches apart on a baking sheet. Top each with 2 Tbsp 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.
