Arrange aubergine slices in layers in a colander, sprinkling each layer with coarse salt. Cover with a plate and place heavy tins on top to press down on the aubergines. Set the colander on a plate and leave them to 'sweat' for 30 minutes and release some of their juices. Rinse well and pat dry with kitchen paper.
Meanwhile, make the tomato sauce. Heat oil in a pan over low heat; add chopped carrot, onion and celery. Cook for a few minutes, or until the onions have softened. Add the tomato passata and continue to cook for about 20 minutes, stirring from time to time. Season to taste and set aside.
Cook the aubergines on both sides under the grill or on a griddle pan over medium heat. Depending on the size of your grill or pan, this could take 15 to 20 minutes to complete. Set the aubergines aside.
Preheat the oven to a 180 C / Gas 4.
Spread a ladle of tomato sauce on the bottom of a large baking dish. Add a layer of grilled aubergines; cover with tomato sauce, grated Parmesan and some of the mozzarella. Add a second layer of aubergines and proceed the same way making as many layers as the ingredients allow. Top the last layer with tomato sauce and grated Parmesan.
Bake in preheated oven for 30 minutes, placing under the grill for the last 5 minutes to brown, if necessary. Allow to stand for a few minutes before serving.
To make ahead
Parmigiana, similar to many Italian baked dishes with tomato sauce, is much tastier if it has had time to stand and soak up the sauce. Prepare up to 2 days ahead, store in the fridge and reheat quickly in the oven before serving.