2 Tbsp olive oil
1 small onion, finely chopped
1 garlic clove, minced
1 14oz can whoole tomatoes in juice
1/2 cup water
1/2 tsp sugar
3Tbsp finely chopped basil
Heat oil in heavy medium sauscepan, 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 tsp salt and simmer until slightly thickened, about 10 minutes. Stir in basil and keep warm.
6 Tbsp olive oil
Cut 12 rounds, 1/3 inch thick. Brush both sides with 2 Tbsp oil and season with 1/2 tsp salt. Bake on an oiled baking sheet, turning once, until golden and tender.
3/4 cup plain dry bread crumbs
1/2 cup finely chopped flat-leaf parsley'1/2 cup grated Parmigiano Reggiano
2 garlic clove, minced, divided
6 large eggs (I used four), lightly beaten
1/2 cup water
Heat 3 Tbsp 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 Tbsp oil to skillet and cook remaining garlic with red-pepper flakes, stirring, until garlic is golden, about 30 seconds. Add 1/2 lb fresh arugula, coarse stems discarded, coarsely chopped. Add 1 cup packed basil leaves, coarsely chopped.
Cook until just wilted.
Now, arrange the 4 egg patties about 3 inches apart on a baking sheet. Top each with 2 Tbsp tomato sauce, 1 slice mozzarella, 1 or two eggplant slices, 2 more Tbsp tomato sauce, another eggplant slice, arugula mixture, and remaining eggplant. Bake until chees melts, 5 to 10 minutes. Drizzle with additional oil and serve remaining sauce on the side.