Slice eggplants and potatoes into ½ inch slices.
Dry and place the potatoes and eggplants in a flat oven tray and brush them with a bit of olive oil and bake them for 10 minutes until golden but not crispy.
In a large sauté pan, brown the beef mince. Add onion and sauté for about 5 minutes. Add garlic and cook for 1 more minute. Add tomato paste, crushed tomatoes and All spice. Allow the sauce to simmer uncovered for 15 minutes so that excess liquid can evaporate. Season to taste with salt and pepper.
Pour the milk in a pot and add the flour in the cold milk. Warm the milk and flour up and continuously whisk the mixture, using a whisk to dilute the flour. Add butter. Simmer over low heat until it thickens a bit but does not boil, stirring continuously. Remove from heat and stir in the beaten egg, and a pinch of nutmeg. Add the cheese and continue to stir for another minute. Return to heat and stir until sauce thickens.
Place the potatoes in a flat tray. Top with a layer of eggplant slices. Add the mince and sprinkle with the grated cheese. Pour the béchamel sauce over the mince, smooth on top with a spatula and sprinkle with remaining grated cheese. Bake in a 180-200°C oven for 45-50 minutes or until béchamel sauce gets a nice golden brown color. Allow to cool for 15–20 minutes before slicing and serving.