In a small bowl toss together the arugula, vinegar, and the 1 teaspoon oil; set aside. In a shallow dish stir together the bread crumbs and Romano cheese. In another shallow dish beat together the egg and milk. In a third shallow dish stir together the flour and salt. Dip the eggplant slices into flour mixture to coat. Dip the slices into egg mixture, then coat both sides with crumb mixture.
In a 12-inch nonstick skillet heat the 1 tablespoon oil over medium heat. Add eggplant slices; cook for 6 to 8 minutes or until lightly browned, turning once. (Add more oil as necessary during cooking.) Top the eggplant with mozzarella cheese; reduce heat to low. Cook, covered, just until cheese begins to melt.
To serve, place the eggplant slices, cheese sides up, on bottom half of bread. Top with the arugula mixture, tomato slices, and top half of bread. Cut into wedges.