Preheat oven to 400 degrees F.
Combine cheese and crab in a small bowl. Divide crab mixture among the upside-down mushroom caps, stuffing the mushrooms. Carefully lay 6 sheets of phyllo dough on top of one another, brushing every other sheet with melted butter. Cut phyllo lengthwise down the center and then cut each half into 3 pieces to make 6 rectangles. Place a mushroom, cheese side up, in the center of each square. Season with salt and pepper. Lift sides of pastry over mushrooms to resemble small bundles and form a "neck," pinching edges together to seal. Repeat the entire procedure 2 more times.
Place on a greased baking sheet and brush with beaten egg. Bake for about 15 minutes until pastry is crisp and golden. Serve hot.