Working one pastry square at a time, brush an egg wash border on all sides. Set a portobello mushroom, top side down, in the center. Fill cap with a portion of the Mushroom Paté, followed by a portion of Caramelized Onions. Fold in each pastry corner to former a packet, sealing the seams, to form a pastry packet. Set, seam side down, on a large oil-coated rimmed baking sheet. Repeat with remaining puff pastry, mushrooms, paté, and caramelized onions. Brush pastry packets with egg wash, and then light score to pastry for a decorative look. Bake until pastry is golden brown, about 25 minutes.