Shred about 10 ounces cooked beef brisket or chuck (about 2 cups). Beat 1 large egg with 1 tablespoon water to blend.
On a lightly floured surface, with a floured rolling pin, roll 1 sheet of thawed frozen puff pastry dough (half of a 3-oz. package) into a 1/16-inch-thick rectangle. Cut into 3-inch squares.
Mound about 1 tablespoon brisket in center of each square. Top with 1/2 teaspoon prepared onion chutney and 1/8 teaspoon fresh thyme leaves; sprinkle with salt and pepper.
Pull corners of pastry over filling and overlap; brush tops with beaten egg to seal. Set on a baking sheet. Repeat with remaining sheet of puff pastry to roll out, cut, fill, and seal more puffs. Set on another baking sheet.
Bake in a 375° oven until golden brown, 16 to 19 minutes. Serve warm.