Clean and wash the aubergines and cut them lengthwise into 0.5-1 cm thick slices. Place the aubergine slices on a board, sprinkle with salt and cover with kitchen paper. Leave to stand.
Wash and peel the potatoes and cut them lengthwise into thick slices (approx. 0.5 cm). Peel onion and garlic. Chop the onion finely, press the garlic through a garlic press.
Pour the tomatoes onto a sieve, collect the juice and chop the tomatoes. Heat 2 tablespoons of oil in a saucepan, sauté the onion and garlic and add the tomato paste. Add tomato juice and pieces, season with salt and pepper and bring to the boil.
Let the sauce thicken for 5-10 minutes. In the meantime, wash the basil, dab dry and cut into fine strips, except for a little garnish. Stir the strips into the sauce, except for a teaspoon.
Dab potato and eggplant slices dry. Heat the remaining oil in one or two large frying pans and fry the slices until golden brown, turning them over. Season with a little salt and pepper and drain on kitchen paper.
Place the aubergine and potato slices and tomato sauce alternately in flakes in a large casserole dish. Sprinkle with cheese and bake in the preheated oven (electric oven: 200 °C/ gas: level 3) for 20-25 minutes.
Sprinkle with remaining basil and serve garnished.