Combine the passata, basil and garlic in a bowl, season with salt and pepper and stir in half the oil.
Toss the aubergines and courgettes in the remaining oil to coat. Heat a ridged griddle pan over a high heat until smoking hot. Add the vegetables, in batches, and cook for 2–3 minutes on each side until tender all the way through.
Spoon a little of the passata mixture on to the base of a deep baking dish, about 30 x 20 cm (12 x 8 inches). Cover with a layer of mixed aubergines and courgettes, then scatter with some of the mozzarella. Spoon over 4 tablespoons of the passata mixture and scatter with Parmesan. Continue layering in this way until all the ingredients are used up, finishing with a layer of passata mixture and Parmesan.
Bake in a preheated oven, 180°C (350°F), Gas Mark 4, for 25–30 minutes, or until golden and bubbly.