Greek-Style Lamb and Eggplant Lasagne

2 pounds eggplant (about 2 medium)
Olive oil for brushing

For meat sauce:
2 large onions, chopped (about 2 cups)
3 garlic cloves, minced
2 tablespoons olive oil
1 1/2 pounds ground lamb (preferably 10 per cent fat)
A 28 to 32 ounce can whole tomatoes with juice
2 tablespoons tomato paste
1 teaspoon cinnamon
1/8 teaspoon ground allspice

For white sauce:
1/2 stick unsalted butter
5 tablespoons all purpose flour
4 cups milk
1 large egg, lightly beaten
An 8-ounce package feta, crumbled (about 1 1/2 cups)
Twelve 7 by 3 1/2-inch sheets dry no-boil lasagne pasta

Cut eggplant lengthwise into 1/3-inch thick slices and sprinkle slices on both sides with salt. arrange eggplant in a large colander and weight it with a heavy bowl. Drain eggplant 1 hour. Remove bowl and pat eggplant dry.
Set broiler rack so that eggplant will be about 4 inches from heat and preheat broiler. Arrange eggplant in one layer in oiled shallow baking pans and brush with oil. Broil eggplant until golden, about 3 minutes on each side.

Make meat sauce: In a large heavy non stick skillet cook onions and garlic in oil over moderate heat, stirring, until softened. Add lamb and cook, stirring to break up, until it is no longer pink, about 3 minutes. Stir in remaining meat sauce ingredients and salt and pepper to taste and simmer, uncovered, 30 minutes. Meat sauce may be made 3 days ahead, cooled completely, and chilled, covered.
Make white sauce: In a heavy saucepan melt butter over moderately low heat and whisk in flour until smooth. Cook roux, whisking, 3 minutes. Whisk in milk in a stream and salt and pepper to taste and simmer, whisking occasionally, 5 minutes, or until thick. Whisk in egg, whisking vigorously, and simmer sauce, whisking, 1 minute. White sauce may be made 2 days ahead and chilled. Its surface covered with plastic wrap.

Preheat oven to 375 degrees F. Pour 1 cup meat sauce into a flameproof baking dish, 13 by 9 by 2 inches (sauce will not cover bottom completely), and cover with 3 lasagne sheets, making sure they do not touch each other. Spread about 1 1/2 cups meat sauce over pasta and arrange 1 layer broiled eggplant on top. Gently spread about 1/2 cup white sauce over eggplant and sprinkle with about 1/3 cup feta. Make 2 more layers in same manner, beginning and ending with pasta. Spread remaining white sauce evenly over top pasta layer, making sure pasta is completely covered, and sprinkle with remaining feta. Cover dish tightly with foil, tenting slightly to prevent foil from touching top layer, and bake in middle of oven 30 minutes. Remove foil and bake lasagne 10 minutes more, or until top is bubbling. Set broiler rack so that lasagne will be about 4 inches form heat and preheat broiler. Broil lasagne 1 to 2 minutes, or until golden. Let lasagne stand 5 minutes before serving. Serves 8 as a main course.