Pour about 1/4 cup extra-virgin olive oil on a rimmed baking sheet. Season the cut eggplant with salt and pepper, to taste, and arrange, cut side down, in the oil. Put in the oven and roast until tender, about 20 minutes.
Peel the remaining eggplant and dice into 1/2-inch pieces. In a large skillet, over medium-high heat, add 1 tablespoon of extra-virgin olive oil. Stir in the beef or lamb, the tomato paste, onions, garlic, diced eggplant and raisins. Season with salt and pepper, to taste.
Add the wine or stock, and cook until the eggplant and onions are soft, about 8 to 10 minutes.
Remove from the heat and stir in the pine nuts and basil.
While the meat cooks, melt the butter in small skillet over medium heat, add the bread crumbs and toast until golden.
Transfer to a bowl and cool.
Stir the parsley and cheese into the bread crumbs and add half of this mixture to the meat.
Remove the eggplant from the oven, flip it over and transfer to a serving platter. Top each halved eggplant with a mound of stuffing and sprinkle the remaining cheesy bread crumbs over the top before serving.
