Drain yogurt in a cheesecloth-lined sieve until thickened, 2 hours or overnight.
Place turkey in a medium saucepan over medium heat; cook until browned, about 6 minutes. Using a slotted spoon, transfer to a medium bowl. Add onion, garlic, cinnamon, salt, nutmeg, and pepper to saucepan; cook until onion is translucent, about 10 minutes. Return turkey to saucepan with tomatoes, tomato paste, and oregano. Bring to a boil; reduce heat to medium low; simmer until sauce has thickened, about 1 hour. Remove from heat; stir in chopped parsley; set aside.
Preheat broiler. While sauce cooks, cut eggplants into 1/4-inch slices. Sprinkle with salt on both sides. Place in a colander over a bowl; let stand 1 hour to drain. Discard liquid; rinse each slice under cold running water to remove all salt and juice. Place slices on several layers of paper towels; press out water. Lay dry slices on a clean baking sheet; coat with olive-oil spray; broil until browned, about 2 minutes. Turn; coat with olive-oil spray; broil until browned, about 2 minutes more. Repeat until all eggplant slices have been broiled; set cooked eggplant aside.
Place drained yogurt in a small bowl. Add Parmesan and eggs. Whisk together briskly with a fork; set aside.
Preheat oven to 400 degrees. Assemble moussaka: Place a layer of eggplant on the bottom of an 8-by-8-inch baking pan. Cover with half the turkey sauce. Place another eggplant layer, then the remaining turkey sauce.Add a final eggplant layer; cover with reserved yogurt mixture. Bake until mixture is bubbling and top starts to brown, about 30 minutes. Transfer to a heat-proof surface; let sit until moussaka cools slightly and firms, about 10 minutes. Cut into squares; serve.