Step 1: Preheat your oven to 400°F (200°C).
Step 2: Line a baking sheet with parchment paper.
Step 3: Roll out the puff pastry to about 1/8 inch thick if not already rolled.
Step 4: Spread the pesto evenly over one sheet of puff pastry.
Step 5: Sprinkle the grated Parmesan cheese over the pesto layer.
Step 6: Fold the second sheet of puff pastry over the first and press gently.
Step 7: Cut the pastry into strips about 1 inch wide and twist each strip.
Step 8: Whisk together the egg and water, and brush over each twisted cheese straw.
Step 9: Sprinkle the remaining Parmesan cheese over the straws.
Step 10: Bake for 15-20 minutes until puffed and golden brown.