1. Preheat broiler to high.
2. Brush eggplant slices with 1 tablespoon oil. Place half of eggplant on a foil-lined baking sheet coated with cooking spray; broil 5 inches from heat for 5 minutes on each side or until browned. Repeat procedure with remaining eggplant. Set eggplant aside.
3. Heat a large skillet over medium-high heat. Add remaining 1 tablespoon oil to pan; swirl to coat. Add chopped onion to pan; sauté 8 minutes. Add garlic; sauté 1 minute. Add bulgur; cook 3 minutes or until bulgur is lightly toasted, stirring frequently. Add ground allspice, cinnamon, and cloves; cook 1 minute, stirring constantly. Stir in vegetable broth, oregano, and tomatoes. Bring to a boil; reduce heat, and simmer 20 minutes or until thickened, stirring occasionally.
4. Melt butter in a saucepan over medium heat. Add flour; cook 1 minute, stirring constantly with a whisk until well blended. Gradually add milk, stirring constantly with a whisk. Bring to a boil; reduce heat to medium-low, and simmer 5 minutes or until thickened, stirring frequently. Stir in cheese and salt. Remove from heat, and cool slightly. Add egg, stirring well with a whisk.
5. Preheat oven to 350°.
6. Arrange half of eggplant in an 11 x 7-inch glass or ceramic baking dish coated with cooking spray. Spread the bulgur mixture evenly over eggplant; arrange remaining eggplant over bulgur mixture. Top with milk mixture. Bake at 350° for 40 minutes, and remove from oven. Increase oven temperature to 475°. Return dish to oven for 4 minutes or until the top is browned. Let stand for 10 minutes before serving.