In a 12-inch frying pan, pour olive oil until it sits 1-inch deep. Heat the oil over medium-high heat until it is hot, but not smoking. Add the eggplant slices and fry, in batches, until lightly browned on each side. Remove the eggplant slices to drain on paper towels

To make the frittata: whisk the eggs and the 1/3 cup Parmesan together in a bowl. Add the parsley, and season the mixture, to taste, with the salt and pepper. Heat 2 tablespoons olive oil in an 8 to 10-inch nonstick frying pan over medium-high heat. Pour in the egg mixture and cook, swirling the egg mixture to cover the bottom of the pan. Cook until the omelet sets and the bottom turns a light golden brown. Invert the frittata over a clean plate and then return to the pan, top-side down, golden brown-side up. Cook for another 1 to 2 minutes until the bottom side is also golden brown. Transfer the frittata to a cutting board and, using a sharp knife, cut into 1 1/2-inch strips. Cut each strip in half, crosswise.

Preheat the oven to 350 degrees F.

Place a frittata strip in the center of each eggplant piece. A few of the longer eggplant slices will need 2 shorter strips of frittata. Roll the stuffed eggplant up from 1 end to the other and place, seam side down, in a lightly greased baking pan. The rolls should be placed side by side, somewhat tightly, together. Spoon the tomato sauce over the eggplant rolls and drizzle lightly with olive oil. Sprinkle lightly with the remaining 2 tablespoons Parmesan. Bake in the preheated oven for 30 minutes. Let rest 5 minutes before serving.