Peel eggplants. Cut eggplants into slices then sprinkle with salt and let stand for 30 minutes. Pat dry with paper towel. This process releases the bitter juices.
Preheat oven to 220 degrees C.
While the eggplants are standing prepare the béchamel sauce. Bring the milk with the garlic cloves to the boil then remove from heat. Set aside.
In a saucepan melt butter and stir in flour. Heat stirring constantly then gradually add the hot milk through a sieve to the saucepan. Continue stirring constantly until a thick béchamel without lumps is formed. Simmer on a very low heat, stirring occasionally for 15 minutes. Stir in the cream and season with salt and pepper. Set aside.
Drizzle the eggplants with olive oil and bake in preheated oven 20 minutes until brown on each side. Crush the boiled garlic cloves and mix in. Stir in the basil and pepper.
Grease a rectangular baking dish. Reheat the béchamel sauce stirring it well. If it is too stiff add some more milk.
Pour some of the bechamel into the bottom of the dish. Lay a layer of lasagne sheets over it and spread some of the eggplant mixture over it. Spread more béchamel over this. Sprinkle mozzarella and Parmesan on top. Repeat the process in the same fashion until all the eggplant and mozzarella are used up (keeping a little bit of the béchamel sauce over).
Place the last layer of lasagne sheets on top then spread with béchamel sauce and sprinkle with remaining Parmesan. Press down gently on the lasagne so that the sheets are completely covered with the sauce.
Cover with aluminium foil and bake for 30 minutes at 200 degrees C. Remove the aluminium foil and bake another 15 minutes. Let stand for at least 10 minutes before serving.