1Heat the oven to 425°F and line two sheet pans with parchment. Lay the eggplant rounds on one pan and the potato slices on the other. Brush both sides with 3 tbsp of the olive oil, sprinkle with salt and pepper, and roast for 25 to 30 minutes, flipping once, until the eggplant is soft and browned and the potatoes are tender at the edges. Lower the oven to 375°F.
2While the vegetables roast, warm the remaining 2 tbsp olive oil in a large skillet over medium-high heat. Add the lamb, break it up, and cook for 8 minutes until browned. Spoon off all but about 2 tbsp of the fat if the pan looks greasy.
3Stir in the onion and cook for 5 minutes until soft. Add the garlic and tomato paste and cook for 1 minute, stirring, until the paste darkens a shade.
4Pour in the red wine and scrape up the browned bits. Simmer for 2 minutes, then add the crushed tomatoes, cinnamon, allspice, oregano, 1 tsp salt and plenty of pepper. Simmer over medium-low heat for 20 minutes, stirring now and then, until thick and spoonable. Stir in the parsley and taste for salt.
5Make the béchamel: melt the butter in a saucepan over medium heat. Whisk in the flour and cook for 1 minute without letting it color. Slowly whisk in the warm milk and keep whisking until it thickens to the consistency of heavy cream, about 6 minutes. Take it off the heat and stir in the nutmeg, half the parmesan, and 1/2 tsp salt. Let it cool for 5 minutes, then whisk in the beaten eggs so they do not scramble.
6Build the moussaka in a 9x13-inch baking dish: lay down all of the potato slices, overlapping them slightly. Cover with half of the eggplant, then spread all of the lamb sauce over it, then finish with the rest of the eggplant.
7Pour the béchamel over the top and spread it to the edges. Scatter the remaining parmesan over it.
8Bake at 375°F for 35 to 40 minutes, until the top is puffed and deeply golden in spots and the edges are bubbling. Rest for 15 minutes before cutting so the layers set and slice cleanly.