Step 1Heat oven to 350 degrees. Line a large baking pan or rimmed cookie sheet with parchment paper.
Step 2In a nonstick skillet, heat 2 tablespoons olive oil over medium heat. Add the onions and cook, stirring frequently, until they are soft and translucent, about 5 minutes. Add the garlic and cook just until the garlic is aromatic, about 1 minute. Remove from heat and place the onions and garlic in a bowl. Stir in the matzo meal, parsley, wine, 2 tablespoons olive oil, 1 teaspoon salt and several grinds of pepper, or to taste.
Step 3Clean the mushrooms with a soft brush or a damp paper towel. Remove the stems with a paring knife. Use a spoon to scrape out the gills. Rub the mushrooms generously with olive oil (the mushrooms will soak up about 1/2 cup of oil), inside and out. Set the mushrooms on the baking pan.
Step 4With a spoon, fill the mushrooms with equal amounts of stuffing. Drizzle the remaining 4 tablespoons or so olive oil evenly over the filled mushrooms.
Step 5Bake for 30 minutes, or until golden brown. If desired, drizzle over a little more oil before serving. Serve warm.