In a large mixing bowl, combine all of the ingredients and stir to combine. Set aside.
Preheat the oven to 350 degrees.
Slice the ends off the eggplants and slice lengthwise into ΒΌ-inch-thick slices. Set out 3 baking sheets or platters. Place the whole-wheat flour, salt, and pepper in one baking sheet, the egg, egg whites, and milk in another, and the breadcrumbs in the last sheet.
Coat all of the eggplant slices first in the flour, then the egg, and lastly cover completely with the breadcrumbs, shaking off the excess at each step. Rub a cast-iron or nonstick skillet or griddle with 1 teaspoon oil and place over medium-high heat. Cook the eggplant in batches without crowding for 3 minutes on each side. Wipe the skillet with paper towels or cloth between batches then rerub with oil each time.
Coat a baking sheet with 2 teaspoons vegetable oil. Transfer the eggplant to the baking sheet in a single layer (you may need multiple baking sheets). Place in the oven and bake for 15 minutes.
Lay a slice of baked eggplant on a flat surface and place about 2 tablespoons of the cheese mixture onto the smaller end of the eggplant. Roll the eggplant and place seam side down in a baking dish where the rolls fit snugly next to each other.
Spoon the tomato sauce over each roll and bake, covered, for 40 minutes. Then, sprinkle with mozzarella cheese and bake, uncovered, for 5 more minutes. Transfer to a plate and garnish with fresh parsley. Enjoy!