Put enough sunflower oil in a deep frying pan to come 2 cm up the sides. Heat the oil to 180°C/350°F, and fry the aubergine slices in a single layer until golden brown on both sides (this may take longer than you think, probably 5 minutes a side). Repeat this operation until all the slices are fried. Spread the slices out on several layers of kitchen paper to absorb the oil that will pour out of them, and put some more on top. Leave for 15 minutes or so.
Make a tomato sauce by cooking the tomatoes, garlic, torn-up basil and 30 ml of the olive oil with seasoning for 10 minutes.
Preheat the oven to 200°C/400°F/Gas 6. Prepare a gratin dish by rubbing it with 15 ml of the olive oil and then a thin layer of the sauce. Cover the sauce with a single layer of aubergine slices (approximately a third of them), then with half the sliced mozzarella. Add another thin layer of tomato sauce, season generously and sprinkle with a third of the Parmesan. Repeat this whole operation again. Top the assemblage with a final layer of aubergine and the remaining Parmesan. Sprinkle with the remaining olive oil and bake until golden brown and bubbling, about 20 minutes.