1. Place mushrooms in a large bowl. Combine oil and wine and pour over mushrooms, then season well with salt and pepper.
2. Combine sauce ingredients in a large bowl and add half of the sautéed onions to the sauce. Place ground beef in another bowl and mix well to break up any lumps. Add 1/2 cup sauce and the rest of the sautéed onions to the meat and mix well to combine. Season with salt and pepper.
3. Preheat oven to 375 degrees Fahrenheit (190 degrees Celsius).
4. Place mushrooms, cap side down, on two lightly greased baking sheets, leaving some space between each of them. Using your hands, form meat into balls to fill each mushroom, and place in the center of the cap. The size of the meatball should be relative to the mushroom it is being placed in. Drizzle each mushroom with a tablespoon of reserved sauce.
5. Place baking sheets in oven and bake for 20 minutes. Remove from oven and drizzle each meatball with additional sauce.
6. Return to oven, alternating the trays on top and bottom oven racks, and bake an additional 15 minutes. Serve immediately.