Preheat oven to 400F. Grease 9 to 10 inch sqaure baking dish. Seperate mushroom caps and stems; set caps aside. Finely chop stem pieces. Melt butter over medium heat.
Add onion; saute until soft, about 3 minutes. In bowl, combine sauteed onion, chopped mushroom stems, bread crumbs, sour cream and hard-cooked egg.
In a bowl, beat egg with fork. Add to onion mixture. Stir in parsley, salt and pepper until blended. Pack about 1 rounded teaspoon stuffing mixture into each reserved mushroom cap. Arrange, stuffing-side up, in greased dish.
Place 1 cheese square on each cap. Bake, uncovered, 20-25 minutes.