Heat the oven to 400 degrees F. Beat the egg and water in a small bowl with a fork or whisk. Place the Parmesan cheese into a shallow dish.
Unfold 1 pastry sheet on a lightly floured surface. Cut the pastry sheet into 6 (5 x 3-inch) rectangles. Repeat with the remaining pastry sheet, making 12 in all.
Place 1 cheese stick on a long edge of each pastry rectangle. Roll up the pastry around the cheese and press the seams and pinch the ends to seal. Brush the tops of the rolls with the egg mixture. Dip the tops into the Parmesan cheese. Place the rolls, seam-side down, onto a baking sheet. Prick the tops of the rolls with a fork.
Bake for 15 minutes or until the sticks are golden brown. Remove the sticks from the baking sheet and let cool on a wire rack for 10 minutes.
Heat the sauce in a 1-quart saucepan over medium heat until hot and bubbling, stirring occasionally. Serve the sauce with the sticks for dipping.