Preheat oven to 400°. Place beef in a small bowl; sprinkle with dressing mix and mix lightly but thoroughly. Shape into eight 1/2-in.-thick patties.
In a large nonstick skillet, cook burgers over medium heat 3-4 minutes on each side or until a thermometer reads 160°. Remove from heat.
Meanwhile, in a small bowl, whisk egg with water. On a lightly floured surface, unfold puff pastry; roll into a 12-in. square. Cut pastry into four 6-in. squares; cut squares in half to make 8 rectangles. Place a burger on 1 end of each rectangle; top with cheese. Brush edges of pastry with egg mixture. Fold pastry over burger to enclose; press edges with a fork to seal.
Transfer to a parchment-lined baking sheet. Brush tops with egg mixture. Bake 15-20 minutes or until golden brown.
1 appetizer: 271 calories, 16 g fat (6 g saturated fat), 54 mg cholesterol, 488 mg sodium, 20 g carbohydrate (0 sugars, 2 g fiber), 11 g protein.