Preheat the oven to 200C/180C Fan/Gas 6.
In a large frying pan, place the lamb, onion, garlic, oregano, mint, bay and cinnamon and fry on medium heat for around 10 minutes, stirring occasionally to break up the meat.
Add in the flour and a pinch of salt and pepper. Add in the wine, chopped tomatoes and tomato puree and bring to a simmer. Cook it for 30 minutes, stir occasionally until the sauce is thickened and the meat is cooked.
Place the aubergine in a colander and sprinkle some salt. Let is sit for 10 minutes. Then, rinse the aubergines and dry them with a towel. Fry the aubergines for 3 minutes on each side. Remove from heat and set aside on a dry kitchen paper.
Cook the potatoes in a boiling water for 5 minutes and drain the water.
For the white sauce, melt the butter in a saucepan and add the flour. Cook for a few seconds and gradually add in the warm milk (previously heat the milk in the microwave for 30 seconds). In the end, add in the Parmesan and nutmeg. Simmer for another 2 minutes and remove from heat. Season with salt and pepper. Once the sauce is cooled, add in the egg.
Take one shallow ovenproof dish and spoon in one-third of the meat. Cover it with one-third of the potatoes and a third of the aubergines (roughly arrange them on top.) Repeat the same for two more layers, finishing with the aubergines. Pour over the white sauce into a thick layer. Sprinkle with some Parmesan.
Bake for 35-45 minutes until it is golden brown and bubbling. Let it cool for 10-15 minutes before serving.