Preheat the oven to 180° C (360° C).
Heat a large frying pan on medium heat and add some vegetable oil in just enough to cover the bottom of the pan.
When the oil is hot enough, add the potatoes in batches and fry until lightly brown on both sides, remove from the pan and set aside on a plate.
Fry the aubergines in the same pan until both sides are lightly browned and they started to soften. Remove from the pan and set aside on a colander to drain excess oil.
To assemble the lamb moussaka, layer half of the aubergines on the bottom of an oven dish then top with half of the mince filling and spread nicely all over the eggplants.
Layer the potatoes on top of the filling, sprinkle on some salt then top with the rest of the mince filling.
Nicely cover with the remaining aubergines and pour over the bechamel sauce. Spread evenly using a silicone spatula or a pallet knife.
Sprinkle evenly the 25 g grated parmesan and 25 g grated cheddar then cover with the breadcrumbs.
Cook in the preheated oven for about an hour or until it is golden brown and bubbling away.
Let it cool down for 15 to 20 minutes before serving.