Heat the oven to 220°C / 200°C fan. Cut the aubergines lengthways into 8 mm slices and peel and slice the potatoes 5 mm thick. Spread over lined baking trays, brush with 50 ml of the olive oil and season with ½ teaspoon of the salt. Roast for 25–35 minutes, turning once, until tender and lightly coloured. Work in batches if needed.

Peel and finely chop the onions and garlic; finely chop the tomatoes. Heat the remaining 20 ml oil in a wide pan. Soften the onions for 8 minutes, then add the lamb and break it up, cooking until browned all over. Stir in the garlic, cinnamon and tomato purée for 1 minute.

Add the tomatoes, 150 ml tap water, ½ teaspoon salt and the pepper. Simmer uncovered for 30–40 minutes, stirring, until the lamb is cooked through and the sauce is thick rather than watery. Add a little water only if needed to prevent sticking.

For the béchamel, melt the butter in a saucepan. Stir in the flour and cook for 2 minutes. Whisk in the milk gradually until smooth. Bring to a gentle bubble and stir for 5–7 minutes until thick. Remove from the heat and stir in the nutmeg, remaining salt and 40 g of the grated Parmesan.

Separate the eggs. Let the sauce cool for 5 minutes, then beat the two yolks into the béchamel, one at a time. The whites are not needed: refrigerate for another use or freeze promptly. Turn the oven down to 190°C / 170°C fan.

In a deep baking dish about 25 × 30 cm, arrange the potatoes, half the aubergines, the lamb sauce and the remaining aubergines. Spread the béchamel over the top and scatter with the remaining 20 g Parmesan. Bake for 40–50 minutes until golden and piping hot right into the middle; a clean probe should reach 75°C for 30 seconds. Rest for 15 minutes before cutting into six portions.