Unfold puff pastry sheet on a lightly floured board; cut into quarters, each 5 inches square. Roll out each slightly into a 6-inch square.
In a small bowl, mix oyster sauce and puréed chipotle chilies. Brush mixture generously over both sides of mushroom caps, using it all.
Center each cap, gill side up, on a puff pastry square. Fold corners of pastry over mushroom to enclose, overlapping slightly; pinch edges together to seal. Set bundles, seams down and slightly apart, on a 12- by 15-inch baking sheet lined with cooking parchment.
In a small bowl, beat egg to blend with 1 tablespoon water. Brush tops and sides of pastry bundles lightly with egg mixture (discard extra). With a small, sharp knife, cut three or four slits (each about 1/4 in. long) through pastry across top of each bundle.
Bake in a 425° regular or convection oven until puff pastry is golden brown, about 10 minutes. Serve warm.
Cleaning mushrooms: Trim off tough or discolored bottoms of mushroom stems and any bruised spots or blemishes. (For shiitakes and oysters, remove the entire fibrous stem.) For firm mushrooms such as portabellas, wipe dirt off with a damp cloth or place in a colander, rinse thoroughly under cool running water, and pat dry with towels. For delicate mushrooms that have lots of places for dirt to hide, such as chanterelles and hedgehogs, submerge in a bowl of cool water and gently agitate with you hands to loose any particles. Drain, rinse carefully under running water, and gently pat dry with a towel.