1. Preheat the oven to 400°F. Grease a baking sheet with brushed olive oil. Carefully remove stems from mushrooms and roughly chop the stems. Place the mushroom caps on the baking sheet.
2. In a medium skillet over medium heat, melt the butter, adding chopped mushroom stems and sauté for a few minutes until all the moisture is dissolved. Fold in the garlic and cook until fragrant, approximately 1-2 minutes. Add the breadcrumbs and let toast slightly, about 2-3 minutes. Season with salt and pepper, to taste. Remove from the heat and allow to cool slightly.
3. In a large bowl, mix together the mushroom stem mixture, the mayonnaise (or cream cheese), parsley and Parmigiano- Reggiano. Season with salt and pepper to taste. Fill the mushroom caps with filling and sprinkle with extra freshly grated Parmigiano-Reggiano.
4. Bake until the mushrooms are soft and the tops are golden-brown, about 20 minutes.
5. Garnish with parsley and serve.