Preheat grill or grill pan to medium-high heat.
Drizzle or brush the eggplant slices with EVOO, season with salt and pepper and grill until tender, 2 minutes each side.
Preheat a large, nonstick skillet over high heat, add about 1/4 cup of EVOO. Once a small amount of smoke starts to form above the surface of the pan, add the ground meat. Break up the meat as it cooks with the back of a spoon or wooden spatula and continue to cook until meat has completely browned.
While the meat is browning, toast the bread. After the bread is toasted, butter and chop it into tiny pieces and reserve.
Add the mushrooms to the skillet with the meat. Cook until golden in color, then add 1 tablespoon butter to the skillet along with the garlic. Let the garlic toast a bit, then add rosemary and season with salt and freshly ground pepper. Add the chopped-up bread and moisten the mixture with 1/2 to 1 cup beef stock to make it moist.
Add the reserved meat back to the pan with the mushrooms and give everything a good stir. Make sure to season everything well. Add the chopped-up bread and moisten the mixture with 1/2 to 1 cup beef stock to make it sticky.
Divide the mushroom-meat mixture among the slices of grilled eggplant. Roll each eggplant slice up around the mixture from the narrower end to fat rounded end. Repeat with all eight slices.
To a medium saucepan melt the remaining 3 tablespoons butter. Add the flour and cook for about 2-3 minutes, then add the tomato paste and cook for 1 more minute. Whisk in the remaining beef stock and cook 5-6 minutes, to allow the stock to thicken. Season with salt and freshly ground pepper and spoon the gravy over the top of the eggplant rollups to serve.