Thinly slice the eggplants lengthwise. Layer the slices in a large colander, sprinkling salt between each layer. Leave for 1 hour to extract the bitter juices. Rinse and pat the slices dry on both sides with paper towels. Coat the eggplant slices lightly with flour.
Preheat the oven to 350 degrees F (180 degrees C) and grease a 32 x 20 cm (13 x 8 inch) shallow casserole or baking tray.
Heat 125 mL (1/2 cup) of the olive oil in a large frying pan. Quickly fry the eggplant in batches over moderately high heat until crisp and golden on both sides. Add more olive oil as needed and drain well on paper towels as you remove each batch from the pan.
Make a slightly overlapping layer of eggplant slices over the base of the dish. Season with pepper. Spoon 4 tablespoons of tomato puree over the eggplant and scatter a few pieces of basil on top. Sprinkle with some mozzarella, followed by some parmesan. Continue with this layering until you have used up all the ingredients.
Bake for 30 minutes. Remove from the oven and allow to cool for 30 minutes before serving.