In a large skillet, brown steaks in oil on both sides. Place a wire rack on a baking sheet. Transfer steaks to wire rack; refrigerate for 1 hour. In the same skillet, saute mushrooms and onions in butter until tender; drain. Stir in the salt and pepper.
On a lightly floured surface, roll pastry into a 13-in. square. Cut into four squares. Place one steak in the center of each square; top with mushroom mixture. Combine egg and water; brush over pastry.
Bring up corners to center and tuck in edges; press to seal. Place on a parchment paper-lined baking sheet. Cover and refrigerate for 1 hour or overnight.
Bake, uncovered, at 400° for 20-25 minutes or until pastry is golden brown and meat reaches desired doneness (for medium-rare, a meat thermometer should read 145°; medium, 160°; well-done, 170°). Yield: 4 servings.