Cut the eggplants into long strips, about 1/4" thick, and soak them for 30 min. in salted water, then dry them very well.
Sprinkle olive oil in a tray laid out with baking paper, arrange the eggplants tightly next to each other, pour on olive oil and sprinkle with salt. Bake in a preheated 428°F (220 °C) oven until they change color - about 15 min.
Do the same with the potatoes - cut them into slices, season with salt and olive oil, bake to a nice crisp.
For the bechamel, first heat the milk with the salt and nutmeg. Beat the eggs in a bowl and set them aside.
Heat the butter and olive oil in a pot, braise the flour for about 30 sec., then, while stirring nonstop with a whisk, pour in the warm milk. Stir until the sauce comes to a boil and remove from the stove. Add the eggs while stirring nonstop.
Assembling the moussaka:
Oil a tray and start arranging the ingredients (after dividing them up into 2) - a layer of potatoes, layer of eggplant, layer of mince, then repeat one more time.
Finally, sprinkle the grated cheese over the mince and pour on the Bechamel. Bake in a preheated 392°F (200 °C) oven with the fan on to a nice crisp - about 40 min.