Preheat the oven to 425 degrees F with the rack in the center.
Place the chilled puff pastry on a lightly floured surface. Roll out the dough until it is about 1/8-inch thick, making a 24 by 26-inch piece. Use a sharp knife to trim the edges of the dough; this will help make the straws uniform as they cook. Brush the dough lightly with the melted butter. Sprinkle the dough with 3/4 cup of the Parmesan, 1 tablespoon of the poppy seeds, and 1 teaspoon of the salt.
With the short side of the dough facing you, fold the dough in half, bringing the top edge down toward you. Brush the surface of the dough with the egg. Sprinkle with the remaining salt, remaining poppy seeds, and 1/2 cup Parmesan. Using a very sharp knife, cut the dough vertically into 1/2-inch-wide strips. Transfer 6 of the strips onto a baking sheet, spacing them evenly apart. Grab each end of a dough strip with your fingers, and carefully stretch and twist the strip in opposite directions. Continue with the remaining strips of dough. (Unbaked dough should be stored in the refrigerator to stay cool. The straws may be twisted, covered with plastic wrap, and transferred on the baking sheets to the refrigerator before baking.)
Bake until golden brown, 10 to 12 minutes. Remove and let cool on the baking sheet for 5 minutes to firm up. Using both hands, transfer the sticks to a wire rack or serving platter.
3 3/4 cups all-purpose flour
1 1/2 teaspoons coarse salt
1 pound (4 sticks) unsalted butter, chilled
1 cup ice water
Combine the flour and salt in a large, chilled stainless-steel bowl. Slice the butter into paper-thin pieces. Using a pastry blender, or working with your fingertips, cut the butter into the flour until the butter is the size of peas, about 1/2-inch in diameter (do not use a
Using a fork, stir in the water gradually, pressing the dough together with your hands as it becomes damp. Turn the dough out onto a well-floured board, and roll it into a rough rectangle with a short side facing you. The dough will be very crumbly. Fold the dough into thirds, and turn it a quarter to the right. Roll the dough into a large rectangle, 9 by 16 inches, and fold into thirds again. This completes the first double turn. Brush away any excess flour. Wrap the dough in plastic wrap, and chill for 30 minutes.
Repeat rolling and folding 2 more times, chilling for 30 minutes after each double turn. With each turn the dough will become smoother and easier to handle. Roll out and bake as instructed in the recipe. The dough can be stored, wrapped well in plastic wrap, in the refrigerator for up to 2 days, or frozen for up to 3 months. To thaw, place in the refrigerator for 24 hours.