Preheat the oven to 350 degrees F.

Heat olive oil in a medium skillet over medium-heat. Add eggplant and saute until tender, about 5 minutes. Drain and set aside.

In a mixing bowl, whisk 1 egg with the ricotta. Stir in the reserved eggplant, asparagus, 1/4 cup of the mozzarella, 1/4 cup Parmesan, oregano, salt, and pepper. Set aside.

Put the chicken breasts in a large plastic bag and pound until very thin and of even thickness. Remove chicken from bag and season with salt and pepper. Spoon a generous dollop of the cheese mixture near the bottom edge of 1 piece of chicken and roll it into a cylinder, around the filling, like a jelly roll. Repeat with remaining chicken.

Beat the remaining eggs in a shallow bowl and put the flour and bread crumbs on separate plates. In a large skillet, melt the butter and shortening. Dredge the chicken rolls-ups in the flour, dip in the beaten eggs, and then coat them with bread crumbs. Place the rolls-ups in the skillet and cook until golden brown on both sides, about 10 minutes.

Spread the tomato sauce in a shallow baking dish and place the roll-ups over the sauce. Sprinkle with the remaining mozzarella and Parmesan. Bake for 30 minutes.

Divide the roll-ups among 4 plates, spooning some sauce over the top. Sprinkle with diced tomato and serve immediately.