Preheat oven to 400 degrees. In a small bowl, whisk together egg and 1 tablespoon water; set egg wash aside. In another small bowl, combine Parmesan, mustard powder, salt, and cayenne; set aside.
On a lightly floured work surface, roll out dough (still folded in thirds) to a 16-by-12-inch rectangle. With short side facing you, cut in half crosswise; brush one side of both halves with egg wash. Sprinkle one egg-washed half with cheese mixture. Sandwich halves together, egg-washed sides facing in; press lightly to adhere. Run a rolling pin over dough to seal.
With short side facing you, cut dough crosswise into 24 strips, each 8 inches long and 1/2 inch wide. Place strips, 1 inch apart, onto two baking sheets; twist strips, pressing down on ends to adhere them to sheets. Bake until cheese straws are puffed and golden, 8 to 10 minutes, rotating sheets halfway through.
To give decorative flair to cheese straws, twist the dough, and flatten the ends so they stick to the baking sheets. This way, the strips won't untwist as they bake.