Preheat the oven to 220°C/425°C/gas 7.
Place the pastry on a lightly floured board. Spread half the Pesto in an even layer all over the pastry. Scatter over half the cheese. Fold the pastry in half, short sides together, and press it together firmly with your hands. Roll it out to more or less the original size.
Spread the pastry with the remaining Pesto and cover with most of the remaining Parmesan (save a bit for sprinkling on top). Fold it in half and roll out again to its original size. Don’t worry too much about getting it exactly the same size; the main thing is to make sure it’s not too thick, and that the Parmesan is pressed into the pastry.
Using a sharp knife, cut the pastry into 1 cm-wide strips. Take each strip and twist it from each end to make a spiral. Place the straws on a baking sheet lined with baking parchment, leaving plenty of room between each straw, as they’ll puff up in size during baking. Brush first with the beaten egg and then sprinkle with the rest of the Parmesan. Bake for 10 to 12 minutes, until crisp and golden. Cool on a wire rack, or serve straight away.