Preheat oven to 400 degrees. Quickly rinse mushrooms in water and dry in salad spinner. Separate stems from mushroom caps. Use a small spoon to carefully scoop out the dark mushroom gills to make more room for the stuffing. Discard gills. Use 2 tablespoons of olive oil to grease 9x13" pan. Chop stems finely and place in the center of a large piece of cheesecloth or a clean towel, and twist and squeeze to extract as much juice as you can. Discard juice. Place remaining olive oil in a skillet and heat over medium heat for a few minutes. Add garlic and mushroom stems and cook, stirring constantly, for 4-5 minutes. Turn off heat and add parsley, ham, bread crumbs and Parmigiano Reggiano. Fill mushroom caps and place in pan. Bake for about 20 minutes until sizzling.