3 1/2 pounds eggplant, unpeeled, cut into 1/2-inch-thick rounds
1/2 cup (about) olive oil
1 large onion, thinly sliced
1 cup finely chopped peeled carrots
1 cup finely chopped celery
4 garlic cloves, minced
12 ounces portobello mushrooms, cut into 1/2-inch pieces
1 teaspoon dried oregano
1/2 teaspoon ground cinnamon
1 28-ounce can crushed tomatoes with added puree
1/4 cup chopped fresh Italian parsley
1 cup grated Parmesan cheese
6 tablespoons (3/4 stick) butter
7 tablespoons all purpose flour
3 1/2 cups whole milk
4 large egg yolks
Cover 2 baking sheets with paper towels. Sprinkle both sides of eggplant rounds with salt. Arrange eggplant in single layer atop towels. Let stand 30 minutes.
Position first rack in bottom third of oven and second rack in top third of oven and preheat to 425°F. Remove eggplant and paper towels from baking sheets. Pat eggplant dry. Oil same baking sheets. Brush both sides of eggplant rounds with 1/4 cup oil. Arrange in single layer on baking sheets. Bake 10 minutes. Turn eggplant and rotate pans in oven. Continue baking until tender, about 15 minutes longer. Cool. Reduce oven temperature to 350°F.
Meanwhile, heat 1/4 cup oil in heavy large skillet over medium-high heat. Add onion, carrots and celery. Sauté until onion is very tender, about 12 minutes. Mix in garlic, then mushrooms. Sauté until juices evaporate, about 10 minutes. Mix in oregano and cinnamon. Add tomatoes and parsley. Cook until mixture is thick, about 10 minutes. Season with salt and pepper.
Lightly oil 13x9x2-inch glass baking dish. Arrange half of eggplant rounds in single layer in dish. Spoon half of tomato mixture evenly over eggplant. Sprinkle with 2 tablespoons cheese. Repeat layering with remaining eggplant, tomato mixture and 2 tablespoons cheese.
Melt butter in heavy medium saucepan over medium heat. Whisk in flour. Stir 2 minutes. Gradually whisk in milk. Simmer until sauce thickens, stirring constantly, about 5 minutes. Whisk in 1/2 cup cheese. Season with salt and pepper. Whisk yolks in large bowl to blend. Gradually whisk in hot sauce. Pour sauce over vegetables in dish. Sprinkle 1/4 cup cheese over sauce. (Can be made 1 day ahead. Cover; chill.)
Bake moussaka until heated through and sauce is golden brown on top, about 45 minutes (or about 55 minutes for refrigerated moussaka). Cool 15 minutes.
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