Add the flour to a shallow dish and season with salt and pepper, to taste. In another shallow dish, combine the eggs and the milk and season with salt and pepper, to taste. Dip the eggplant slices into flour, and then into the egg mixture.
For the stuffing:
In a large bowl combine the 3 cheeses, parsley and the 2 beaten eggs. Mix well and season with salt and pepper, to taste. The mixture should be a little wet, so add enough bread crumbs to make a dry mixture that will not stick to your hands.
Lay the eggplant slices on a flat surface and divide the filling between the slices. Roll up and arrange in a lightly greased ovenproof dish. Refrigerate for 1 hour.
Preheat the oven to 350 degrees F.
Stir in the chopped tomatoes, tomato paste and vegetable stock and bring to a boil. Cook for 5 minutes, then reduce the heat and continue to cook at a simmer for another 15 minutes. Add the basil and the butter and adjust the seasoning with salt and pepper, to taste.
To finish the dish:
Remove the eggplant from the refrigerator, top with the fresh tomato sauce and bake until the eggplant is cooked through and the sauce is bubbling, about 20 minutes.