Trim and peel the eggplants, then cut crosswise into ½-inch-thick slices. Sprinkle the eggplant slices on both sides with salt and place them in a single layer on paper towels to absorb the moisture. Let the eggplant slices sit while you cook the lamb mixture.
In a large sauté pan, heat about 1 tablespoon olive oil over medium heat. Add the onions and garlic and cook, stirring, until soft, 3 to 5 minutes. Add the lamb, raise the heat slightly, and cook, stirring and breaking up the meat with a wooden spoon, until browned, about 5 minutes. Add the cinnamon, nutmeg, parsley, chives, tarragon, tomato sauce, and wine and stir well. Adjust the heat to maintain a steady simmer and cook, stirring occasionally, until most of the moisture has evaporated and the flavors are well blended, about 20 minutes. Season with salt and pepper and remove from the heat.
Wipe the eggplant slices dry with paper towels. Brush a large frying pan with olive oil and place over high heat. Working in batches, fry the eggplant slices, turning them and adding more oil to the pan as needed, until browned on both sides and just tender, 2 to 3 minutes on each side. Transfer to paper towels to drain.
Preheat the oven to 350°F. Oil a 9-by-13-by-2-inch baking dish or similar-size gratin dish.
To make the béchamel, pour the milk into a saucepan and heat over medium heat just until small bubbles appear around the edge of the pan. While the milk is heating, in a heavy saucepan, melt the butter over medium heat. Whisk in the flour and continue to whisk until the mixture is bubbly and smooth, about 3 minutes. Do not allow it to color. Whisk the hot milk, 1 cup at a time, into the flour mixture, whisking until smooth after each addition.
Turn the heat to medium-low and continue to cook, stirring constantly to prevent scorching, until nicely thickened, 2 to 3 minutes.
If using the egg yolks, in a medium bowl, whisk the yolks until blended. Slowly ladle about 1 cup of the hot béchamel into the eggs while whisking constantly until fully blended. Then slowly pour the egg mixture into the saucepan while whisking continuously to combine. Stir in the salt and nutmeg, remove from the heat, and keep warm.
Arrange half of the eggplant slices on the bottom of the prepared baking dish. Top with the meat mixture, spreading it evenly, and sprinkle with one-third of the cheese. Arrange the remaining eggplant slices on top and sprinkle with half of the remaining cheese. Pour the béchamel evenly over the surface and sprinkle with the remaining cheese.
Bake until bubbling is visible along the edges and the top is lightly browned, about 1 hour. Remove from the oven and let cool for 20 to 30 minutes, then cut into 3-inch squares to serve.
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