Preheat oven to 375°F.
Remove the mushroom stems and finely chop the stems. But wait! Don't toss the caps!
In a large nonstick skillet, heat 3 Tablespoons of the oil over medium-high heat.
Add the chopped mushroom stems and season with salt to your liking. Stir and cook until the mushrooms have released their liquid and the mixture begins to dry out, about 5 minutes.
In a steady stream, add the Marsala wine and cook until it evaporates completely, about 2 minutes.
Remove skillet from heat and stir in bread crumbs. Allow to cool slightly.
Add the prosciutto, parmesan, parsley, garlic and pepper to taste, to cooled mixture. Stir until fully combined.
On a greased baking sheet place the mushroom caps open side up in one layer.
Spoon the cooked mushroom mixture into the caps. When all the cooked mixture is divided between the caps and slightly mounded, drizzle with the remaining olive oil.
Bake for 30-45 minutes, or until golden. Serve warm.