Start with the tomato sauce. Chop shallots and garlic and sauté them in olive oil in a pot until they soften and get a bit of color. Chop the fresh tomatoes and add them. Cook until the liquid from the tomatoes has almost evaporated. Add sherry vinegar and let it cook off. Add the canned cherry tomatoes and mash them slightly into the sauce. Let it simmer gently for about 20 minutes. Blend the sauce smooth with an immersion blender or in a blender. Season with salt, pepper, and sugar. Let it reduce a bit if it seems too thin.
Pound the chicken breasts between two freezer bags. They should not be completely smashed. Season them with salt and pepper and coat them in a bit of flour mixed with salt and pepper. Then dip them in beaten egg and finally coat them with breadcrumbs mixed with grated Parmesan.
Fry them in plenty of olive oil until they are golden and crispy. In a roasting pan that fits the fried chicken breasts, spread a thin layer of tomato sauce. Place the breasts side by side. Spread a large spoonful of tomato sauce over each breast and sprinkle with grated, firm mozzarella on top. Sprinkle with extra grated Parmesan and put the dish in the oven at 200 degrees Celsius (392 degrees Fahrenheit) until the cheese bubbles and starts to turn golden (about 15 minutes). Serve piping hot with a good mixed salad. If there are extra hungry people at the table, some pasta tossed with extra tomato sauce can also be good.