Unfold the puff pastry sheets on a lightly floured surface. Beat the egg with 1 tablespoon water; generously brush on the pastry sheets (reserve the remaining egg wash).
Combine the cheddar, parmesan, 1/2 teaspoon salt and 1/4 teaspoon pepper in a medium bowl. Sprinkle the cheese mixture over 1 pastry sheet, then place the other pastry sheet on top, egg wash-side down, lining up the edges. Lightly dust the pastry with flour; firmly press the top sheet down, then roll out the sandwiched pastry sheets into a 10-by-12-inch rectangle.
Line 2 baking sheets with parchment paper. Cut the puff pastry rectangle in half lengthwise. Cut each half crosswise into 18 strips. Twist each strip a few times and arrange 1/2 inch apart on the prepared baking sheets. Brush with the reserved egg wash. Refrigerate at least 20 minutes or up to 1 day.
Position racks in the upper and lower thirds of the oven and preheat to 375 degrees F. Bake the cheese straws, switching and rotating the baking sheets halfway through, until puffed and golden brown, 30 to 35 minutes. Slide the straws (on the parchment) onto a rack and let cool completely.