Line a baking sheet with parchment.
Lightly roll out the sheet of pastry into a 10-by-12-inch rectangle, then brush with the melted butter. Sprinkle over the Cheddar, cayenne and 1/2 cup of the Parmesan. Using the rolling pin, press the cheese into the pastry. Cut the pastry into 12 strips. Twist each strip about 5 times and transfer to the prepared baking sheet. Place in the freezer for 20 minutes.
Preheat the oven to 400 degrees F.
Mix together the egg and 1 teaspoon water in a small bowl, brush the strips with the mixture and sprinkle over the remaining 1/4 cup Parmesan. Bake until golden brown and puffed, 10 to 12 minutes.