Using a meat tenderizer or a small, heavy-bottomed saucepan, pound the turkey until 1/8 to 1/4-inch thick.
Combine the flour, 1/2 teaspoon salt and 1/2 teaspoon pepper in a medium bowl. Place the eggs in a second medium bowl. Place the bread crumbs in a third medium bowl. Season the turkey on both sides with salt and pepper Dredge the turkey in the flour and shake off any excess flour. Dip the turkey into the eggs and then into the bread crumbs to coat. Heat the oil in a large skillet over medium heat. Cook the turkey, in batches, until golden brown and cooked through, 2 to 3 minutes on each side, adding more oil, if needed. Remove from the skillet and drain on paper towels.
Arrange the arugula on a large serving platter and top with the tomatoes. Drizzle with olive oil and season with salt and pepper, to taste. Arrange the turkey on top of the tomatoes. Using a vegetable peeler, shave the Parmesan over the turkey. Serve with lemon wedges on the side.