Preheat the oven to 400°F (200°C). Line two baking sheets with parchment paper and set aside.

Unfold both thawed puff pastry sheets onto a large piece of parchment or wax paper so each sheet lies flat.

Cut each pastry sheet into 4 equal long rectangles (cut each sheet into quarters along its longer side). You will have 8 rectangles total.

Spread the 3 tablespoons pesto evenly over 4 of the rectangles (one quarter of each sheet).

Sprinkle the 1/3 cup grated Parmesan evenly over the pesto-covered rectangles.

Place the remaining 4 plain rectangles directly on top of the pesto-and-Parmesan rectangles, aligning edges, and gently press around the edges to seal.

Using a sharp knife or pizza cutter, cut each stacked rectangle into 4 long strips lengthwise so you end up with 16 two-layer strips.

Hold each strip at both ends and twist in opposite directions several times to make a spiral straw; transfer the twisted straws to the prepared baking sheets, spacing them so they are not touching.

In a small bowl, beat the egg with the 1 tablespoon water. Lightly brush the tops of the pastry straws with the egg wash.

Sprinkle the reserved 3 tablespoons Parmesan evenly over the egg-washed straws.

Bake for 12 to 15 minutes, rotating the pans halfway through baking, until the straws are puffed and golden brown.

Remove from the oven and let the straws cool on the baking sheets. Transfer to a platter and garnish with extra Parmesan cheese, if desired. Serve.