Preheat the oven to 350°F (175°C).
Remove the stems from the mushrooms and wash them. Pat the mushrooms dry with paper towels.
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. Allow mixture to cool and transfer it to a mixing bowl.
Stir in wine, Italian-style seasoned breadcrumbs, Parmesan cheese, lemon juice, and mozzarella or fontina cheese. Mix until well combined. Set aside while you arrange the mushrooms.
Brush the mushrooms with olive oil and place them in the cups of a mini muffin tray.
Use a small spoon or cookie scoop to fill the mushroom hollows with the prepared mixture.
Bake in the preheated oven for 8-10 minutes or until the stuffing starts to brown on top and the mushrooms are cooked through.
Remove from the oven. Garnish with parsley, basil, or additional Parmesan cheese if desired.
Serve hot as an appetizer or snack.