Heat the oven to 425°F (220°C). Lightly oil a deep 9-by-13-inch baking dish. Spread eggplant and potato slices on separate rimmed baking sheets. Brush with 3 tablespoons of the oil and season with the vegetable-layer salt and pepper.

Roast for 25–35 minutes, turning the slices and rotating the trays halfway through, until the potatoes are tender and the eggplant is softened with golden edges. Remove any pieces that finish early. Reduce the oven to 375°F (190°C).

Meanwhile, heat the remaining tablespoon of oil in a large skillet over medium-high heat. Cook the onion for about 5 minutes. Add the beef and break it into small pieces; cook for 7–9 minutes, until browned and at least 160°F (71°C). Add the garlic and tomato paste and stir for 1 minute.

Add the crushed tomatoes, water, oregano, cinnamon and beef-sauce salt. Simmer uncovered for 15–20 minutes, stirring, until thick enough that a spoon drawn across the pan leaves a brief clear trail. The sauce should be moist but not watery.

For the béchamel, melt the butter in a saucepan over medium heat. Whisk in the flour and cook for 1–2 minutes without browning. Add warm milk gradually, whisking smooth after each addition. Simmer gently for 5–8 minutes, whisking, until the sauce thickly coats a spoon.

Remove the sauce from the heat. Stir in 40 g of the Parmesan, nutmeg and béchamel salt. Let cool for 5 minutes. Whisk a few spoonfuls of the warm sauce into the beaten eggs, then whisk that mixture back into the saucepan off the heat.

Arrange the potatoes in the baking dish. Layer with half the eggplant, half the beef sauce, the remaining eggplant and remaining beef sauce. Spread the béchamel across the top and sprinkle with the remaining 20 g Parmesan.

Bake for 35–45 minutes, until the top is golden, the edges bubble and the center reaches 165°F (74°C). If it browns too quickly, tent loosely with foil. Rest for 20 minutes before cutting into eight portions.

Refrigerate leftovers in shallow covered containers within 2 hours, or 1 hour above 90°F (32°C). Keep at 40°F (4°C) or below for 3–4 days. Reheat portions to 165°F (74°C).