Heat the oven to 425°F. Place the beef into a lightly greased roasting pan. Season with the black pepper, if desired. Roast for 30 minutes or until an instant-read thermometer inserted into the beef reads 130°F. Cover the pan and refrigerate for 1 hour.
Reheat the oven to 425°F. Beat the egg and water in a small bowl with a fork.
Heat the butter in a 10-inch skillet over medium-high heat. Add the mushrooms and onion and cook until the mushrooms are tender and all the liquid is evaporated, stirring often.
Sprinkle the work surface with the flour. Unfold the pastry sheet on the work surface. Roll the pastry sheet into a rectangle 4 inches longer and 6 inches wider than the beef. Brush the pastry sheet with the egg mixture. Spoon the mushroom mixture onto the pastry sheet to within 1-inch of the edge. Place the beef in the center of the mushroom mixture. Fold the pastry over the beef and press to seal. Place seam-side down onto a baking sheet. Tuck the ends under to seal. Brush the pastry with the egg mixture.
Bake for 25 minutes or until the pastry is golden brown and an instant-read thermometer inserted into the beef reads 140°F.