Lay 1 sheet of the puff pastry on the work surface. Spread the anchovies over it and sprinkle with the Parmesan.
Brush the top of the other sheet of pastry with a little beaten egg. Place this on top of the anchovy and cheese mixture, egg wash-side down, to make a puff pastry sandwich. Roll over the puff pastry sandwich with a rolling pin to seal the 2 sheets together. It should be about the same thickness as 1 of the original sheets of pastry. Brush the top of the sandwich with a little beaten egg and sprinkle with a few black sesame seeds.
Cut the sandwich into 10 x 1.5 cm (4 x ¾ inch) strips and place on a nonstick baking sheet, leaving room between them to allow them to expand.
Place the straws in a preheated oven, 200°C (400°F), Gas Mark 6, and bake for 10 minutes, or until risen and golden brown. Remove from the oven and cool on a wire rack.