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The moussaka can be made up to 2 days ahead. Allow an extra 15-20 minutes if cooking from chilled. The meat sauce can be frozen.
Boil the whole unpeeled potatoes for 25-30 minutes until tender. Drain and set aside until cool enough to handle.
Meanwhile, cut the aubergine into 1cm slices, sprinkle with 1⁄2 tsp salt and leave in a colander over a bowl for 10-15 minutes. Preheat the oven to 190°C, fan 170°C, gas 5.
Pat the aubergine dry on kitchen paper, brush with the oil on both sides and cook on a lined baking tray for 20-25 minutes until golden, turning halfway.
Heat a casserole on the hob and add the mince and onions (no need for oil). Cook over a high heat, stirring well to break up the mince, until the meat starts to brown. Add the garlic, mint, cinnamon and tomato purée and cook for 1 minute. Tip in the tomatoes and add 200ml water. Season and simmer uncovered for 15 minutes.
Scrape the skin off the potatoes and slice thickly. Spoon a third of the meat sauce into a baking dish, then add half the potatoes. Add another layer each of meat sauce and potatoes, then the rest of the meat sauce, followed by the aubergine slices.
Melt the butter in a saucepan, add the flour and cook for 1 minute. Take off the heat and gradually add the milk, stirring continuously. Return to the heat and bring to a simmer, stirring, until thickened. Simmer for 2 minutes then take off the heat, add nutmeg and seasoning to taste, followed by the egg. Pour over the aubergine. Scatter with the Parmesan then bake in the oven for 40-45 minutes until the top is slightly puffy and golden brown. Leave to settle for 10 minutes before serving.