Position 2 oven racks in the center of the oven, and preheat to 400 degrees F. Line 2 baking sheets with parchment.
Roll out 1 puff pastry sheet with a rolling pin on a lightly floured work surface to a 12-inch square.
Whisk the egg in a small bowl with 1 tablespoon water until blended.
Brush the dough with a thin layer of the egg wash. Sprinkle half the Asiago onto the dough, and press it in with the rolling pin or the back of a measuring cup. Cut the dough in half horizontally (a pizza roller is very handy for this). Place one half on top of the other, cheese-side up.
Cut the dough into 1/2-inch vertical strips; you should have 24. Pick up 1 strip with a hand on each end, and twist in opposite directions until you have made a spiral. Place the spiral on the baking sheet, and gently pinch the ends to adhere them to the parchment. Continue with the remaining strips, placing them about an inch apart, 12 per baking sheet. Brush the twisted dough with the egg wash, and bake until golden and puffy, 15 to 17 minutes, rotating the baking sheets halfway through. Let cool completely on the baking sheet. Repeat the process with the remaining puff pastry and cheese to make 24 more cheese straws. Store in an airtight container at room temperature for up to 3 days.
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