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1. 01Roast the vegetables.Heat the oven to 425°F / 220°C. Spread eggplant and potato slices over separate lined trays. Brush with about three-fifths of the oil and season with about one-third of the salt and half the pepper. Roast for 25–35 minutes, turning once, until tender and lightly browned. Swap tray positions if needed.
2. 02Make the meat sauce.Meanwhile, heat the remaining oil in a wide saucepan. Cook onion for 5 minutes, then add beef and break it into small pieces. Cook for 7–9 minutes until browned and at least 160°F / 71°C. Add garlic, tomato paste, cinnamon, oregano, and bay leaf and stir for 1 minute. Add tomatoes, water, and another third of the salt. Simmer uncovered for 15–20 minutes until thick; remove the bay leaf.
3. 03Prepare the béchamel.Melt butter in a saucepan over medium-low heat. Stir in flour for 1 minute. Gradually whisk in warm milk, then simmer gently for 5–7 minutes, stirring, until it coats a spoon. Remove from heat and let stand for 5 minutes. Stir a little warm sauce into the beaten egg, then whisk this mixture back into the pan with cheese, nutmeg, remaining salt, and remaining pepper.
4. 04Arrange the layers.Lower the oven to 375°F / 190°C. Put the potatoes in the bottom of the baking dish, overlapping slightly. Add half the eggplant, spread over the meat sauce, and cover with the remaining eggplant. Spoon the béchamel evenly over the top.
5. 05Bake until golden.Bake for 40–50 minutes, until the top has golden patches and the center reaches 165°F / 74°C. If the top colors too quickly, cover loosely with foil.
6. 06Give it a rest.Rest for 30 minutes before cutting into portions. Lift out with a broad spatula. Refrigerate leftovers within 2 hours of leaving the oven.

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