Heat the oven to 400°F. Beat the egg and water in a small bowl with a fork or whisk.
Heat the oil in a 10-inch skillet over medium heat. Add the broccoli, mushrooms, onion and garlic and cook until the mushrooms are tender and the liquid is evaporated. Remove the skillet from the heat. Add the cheese and stir until it's melted.
Unfold 1 pastry sheet on a lightly floured surface. Roll the pastry sheet into a 12-inch square. Cut into 16 (3-inch) squares. Repeat with the remaining pastry sheet, making 32 in all.
Place 2 teaspoon broccoli mixture in the center of each square. Brush the edges of the squares with the egg mixture. Fold the corners to the center over the filling and press to seal. Place the filled pastries onto baking sheets. Brush with the egg mixture.
Bake for 15 minutes or until the pastries are golden brown. Remove the pastries from the baking sheets and let cool on wire racks for 10 minutes.
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