Dice the union and carrot. Put a large pan on the heat and fry in some oil/butter the carrot and union. After about 5 minutes add the pressed garlic. Fry the garlic together with the union and carrot for another 5-7 minutes. Then add the peeled tomatoes. Make sure to cut them with a spatula, since they are whole, and it is a rustic sauce, you don't want to large pieces. When the juices are bubbeling, add sugar, salt, pepper and the herbs. Start off with little amounts, you can always add more along the way. So season it after your own taste, and when you think it is tasty the sauce is finished.
Slice the aubergines in thin slices ( aprox. 2mm thick ). Layer them in a large oventray and cover each layer with salt. This takes the bitterness and the excess moist out of the slices. Let the slices bath for 20 minutes. After that, take some papertowel/kitchentowel and pat them dry. Brush the grillpan with some oil and grill the aubergines until done. This takes 1-2 minutes on each side. Remember to brush your grillpan with oil everytime you lay new slices on it.
Slice the 2 mozzarella's. Put half of the sauce into the large oventray. On top of the sauce layer half of the aubergines. Cover up with 1 mozzarella and the grated cheeses. Layer the rest of the aubergine slices on top of that and cover that up with the other mozzarella and the grated cheeses. Spoon the rest of the sauce over the cheeses. If there's any cheese left, sprinkle it on top of the sauce. Put the tray in a pre-heated oven ( 175 C ) and let it gratinate for 20 minutes.
Serve with fresh ciabatta bread or some toasted bread.