Slice the eggplant lengthwise and cut into 1/4-inch thick slices, using a mandoline or sharp knife.
Heat a large skillet over medium-high heat and add the oil.
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.
Fry the eggplant, in batches, in the hot oil until golden brown on both sides.
Remove the eggplant to paper towels to drain. Repeat until all the eggplant has been fried.
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.
Add the canola oil to a large saucepan over medium-high heat and add the garlic and onions and saute until golden brown.
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.
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.
