Heat oven to 425 degrees F. Coat 11 x 7 x 2-inch pan with nonstick vegetable cooking spray.
Brush the eggplant with 3 tablespoons oil. Place in single layer on 2 baking sheets.
Bake in 425 degree F oven for 30 minutes, turning over halfway through.
Heat 1 tablespoon oil in large nonstick skillet over medium heat. Add onion; saute 6 to 8 minutes or until softened. Add beef and turkey; cook, breaking up meat with wooden spoon, 6 minutes or until no longer pink. Stir in tomato paste, oregano, pepper, cinnamon, 1 teaspoon salt.
Arrange one layer of eggplant over bottom of prepared pan. Top with meat mixture. Repeat layering and finish with top layer of eggplant.
Shake together 2/3 cup milk and flour in jar with tight-fitting lid to dissolve flour. Combine with remaining milk and salt in saucepan. Bring to simmering, whisking.
Beat egg and egg white in small bowl. Whisk some hot milk into egg mixture; then stir into hot milk mixture in saucepan. Cook, whisking, until thick; do not let boil. Pour over eggplant. Sprinkle with Parmesan.
Bake in 425 degree F oven for 30 minutes or until top is bubbly. Makes 8 servings.