Preparation

• Heat 1 Tbs. olive oil in a 3-quart saucepan over medium heat. Add the onion and garlic and cook until soft and slightly browned, about 3 minutes. Add the tomatoes, 1/4 tsp. salt, and a grind of pepper and simmer, stirring frequently, until the tomatoes cook down to a dry sauce, 20 to 25 minutes, reducing the heat to medium low if necessary. Off the heat, stir in the olives, capers, 1/2 Tbs. olive oil, and salt and pepper to taste.
• Heat 1 Tbs. each olive oil and vegetable oil in a 12-inch skillet over high heat. Add the eggplant and cook, stirring occasionally, until tender and well browned on several sides, 3 to 5 minutes. Transfer to a bowl and cool to room temperature.
• To the eggplant, add the finely grated Parmigiano, 1 Tbs. of the pine nuts, 1 Tbs. lemon juice, the basil, and about half of the tomato sauce. Season to taste with salt and pepper.
• Using a mandoline, slice the zucchini lengthwise about 1/8 inch thick. Select the 24 widest, longest slices and arrange them in a single layer on paper towels. Sprinkle lightly with salt and let sit until pliable, about 3 minutes—you can shingle the layers of zucchini between paper towels to save space. Pat dry. Arrange 3 slices of zucchini on a work surface, overlapping them lengthwise. Spread a heaping tablespoon of the eggplant mixture near one end of the zucchini ribbons and roll the zucchini around the filling to make a roll. Set aside, seam side down, and repeat with the remaining ingredients to make 8 rolls total. You may not need all the filling.
• Put the panko, flour, and eggs in 3 shallow bowls. Lightly coat each roll in the flour, then dip it in the eggs, and coat in the breadcrumbs—it’s fine if it isn’t perfectly coated.
• Heat the 1/2 cup olive oil and 1/2 cup vegetable oil in a 10-inch straight-sided sauté pan over medium heat. Working in two batches, fry the rolls until golden brown on all sides, 2 to 3 minutes per side. As each batch finishes, transfer to a paper-towel-lined plate and sprinkle lightly with salt.
• While the rolls cook, reheat the remaining sauce in a small saucepan, adding about 1/4 cup water, or enough to thin to a wet sauce.
• Whisk the remaining 2 Tbs. olive oil and 1 Tbs. lemon juice in a small bowl. Toss the arugula with the dressing and season to taste with salt and pepper. Serve the rolls topped with the sauce, remaining pine nuts, and shaved Parmigiano, with the salad on the side.