Preheat oven to 400F. Beat egg and water with wire whisk until well blended; set aside. Unfold pastry on lightly floured surface.
Roll into 14x10-inch rectangle; cut in half lengthwise.
Brush both pastry halves lightly with some of the egg mixture.
Sprinkle 1/4 cup of the cheese and 1/4 tsp. of the pepper evenly over one of the pastry pieces; cover with remaining pastry piece, egg-side down.
Roll gently with rolling pin to seal two pastry pieces together.
Cut crosswise into 18 (3/4-inch-wide) strips. Twist each strip, then place, 2 inches apart, on greased baking sheets, pressing down ends of pastry strips to baking sheet to secure.
Brush with remaining egg mixture; sprinkle evenly with remaining 1/4 cup cheese and remaining 1/4 tsp. pepper.