Wash the eggplants and do not trim. Keep them whole and intact. Rub the grill down with some canola oil and place the eggplants on the grill. Cook, turning every 10 minutes, until tender and a toothpick pierces the skin easily, about 40 minutes. (For the oven, place the eggplant directly on the rack and put a baking pan below to catch the drips.)
Set the eggplant on a plate to cool. As soon as it is cool enough to handle, peel the skin away and discard. Cut the eggplant into several large pieces and place in a colander over a deep dish for about 15 minutes. Allow the juices to drain away, squeezing the pieces as necessary to encourage the eggplant to shed it's liquid.
Chop the eggplant very fine and combine in the mixing bowl with the bread crumbs, parsley, garlic, egg, Parmesan, salt, and pepper. Mix it all together with your hands until it's uniform. Shape into little cakes about 2 inches (about 5 cm) across and 1/2-inch thick (about 1 cm). Lay the cakes on a large plate.
Pour the canola oil into the frying pan until it comes about 1/2-inch (1 cm) up the sides of the pan. Turn the heat on high. When the oil is very hot (375° F/190° C), dip the cakes in flour on each side and gently slide into the oil. Don't overcrowd the pan, allow plenty of room between the cakes. When the crust gets nice and brown, flip them over. Use a slotted spoon or spatula to remove to a large plate lined with paper towels. Sprinkle some more salt on top while still hot.
Serve hot or warm with some pasta sauce and sauteed onions (onion confit) on the side.
Baked with mozzarella: Top each fried cake with a slice of fresh mozzarella and bake at 400° F/200° C until the cheese melts.