1. Preheat oven to 425 degrees. Line 2 baking sheets with parchment.

2. On a lightly floured surface with a lightly floured rolling pin, roll puff pastry into a 12-by-16-inch rectangle. With the long side facing you, lightly brush bottom half of rectangle with some of the beaten egg. Sprinkle 1/4 cup Parmesan and the salt on top half. Fold bottom half over top half. Gently roll dough to seal.

3. Cut rectangle lengthwise into thirds. (You should have three 2-by-16-inch strips.) Sprinkle remaining 1 tablespoon Parmesan on 1 strip, poppy seeds on another, and cayenne on remaining strip. Cut each strip lengthwise into eight 1/4-by-16-inch strips.

4. Transfer strips to baking sheets, spacing 1 inch apart. Working with 1 strip at a time, hold each end with your fingers, and carefully twist strip in opposite directions, forming a spiral. Press ends of strips against parchment to prevent unraveling. Freeze or refrigerate until strips are very firm and cold, about 15 minutes.

5. Bake 1 sheet at a time, rotating halfway through, until straws are golden brown and cooked through, 12 to 14 minutes. Transfer straws to wire racks, and let cool.