Preheat the oven to 350°F (175°C).
Remove the stems from the mushrooms and wash them. Pat the mushrooms dry.
Brush the mushrooms with olive oil and place them in a single layer in a baking pan.
In a heavy skillet, heat the remaining olive oil and melt the butter. Add minced garlic and green onions, sauté for 2-3 minutes until fragrant.
Add the mushroom stems to the skillet and cook for 1 minute. Remove from heat.
Stir in wine, Italian-style seasoned bread crumbs, Parmesan cheese, and mozzarella or fontina cheese. Mix until well combined. If the mixture is too dry, add more wine until it sticks together.
Use a cookie scoop to fill the mushroom hollows with the prepared mixture.
Top each mushroom with additional Parmesan cheese and pour a bit of wine over them.
Bake in the preheated oven for 8-10 minutes or until the cheese melts and the mushrooms are cooked through.
Remove from the oven, sprinkle with salt, pepper, and chopped parsley.
Serve hot as an appetizer or snack.