When twisting the straws into shape, don't worry if you lose up to three tablespoons of cheese; plenty will remain inside and you can sprinkle any lost over the outside.
On a lightly floured surface, using a rolling pin, gently roll pastry dough into a 14-inch square 1/8 inch thick. Be careful not to press too hard on the edges. Using a pastry brush, dust off excess flour; brush beaten egg over lower half of dough until it is well moistened.
In a small bowl, mix together salt and cayenne. Sprinkle half the salt mixture over egg-washed portion of dough. Sprinkle half the grated cheese over the same half; fold the uncovered dough over the cheese-covered section. Gently press all over the dough with your fingers to seal the dough halves together.
Brush top of dough again with beaten egg to moisten; sprinkle remaining salt mixture and cheese evenly over the top.
Using a pastry wheel or sharp knife, trim edges, and cut dough into 1/2-inch vertical strips. Twist strips into spiraled straws, and lay 1 inch apart on three ungreased baking sheets. To prevent straws from straightening out again, gently press both ends to adhere them to the baking sheet. Cover with plastic wrap; place in refrigerator 1 hour.
Preheat oven to 425 degrees. Place in oven, and bake 15 minutes. Reduce oven heat to 350 degrees; continue baking until straws are golden brown and cooked all the way through, about 5 minutes. Remove from oven; using a thin spatula, immediately transfer straws to a wire rack to cool completely before serving.