Preheat oven to 375°. Flour a work surface and a rolling pin. Set puff pastry on surface, sprinkle lightly with flour, and roll out into a rectangle about 8 in. by 12 in. and 1/8 in. thick. Arrange pastry so that it's horizontal and trim edges even.
Cover left half of pastry with 1 layer of prosciutto. Using a rasp or box grater, finely grate plenty of parmesan onto prosciutto. Brush right half of pastry with egg wash and fold over prosciutto and parmesan, like closing a book. Sprinkle top with flour and roll back out into a rectangle almost as large as the one you began with. Cover half of pastry with another layer of prosciutto slices and grated parmesan and brush other half with egg wash. Fold egged half over and roll with the pin a few times to seal.
With a sharp knife, cut pastry lengthwise into 1/3-in.-wide strips. Twist each strip loosely a couple of times and put on an ungreased baking sheet (push ends onto sheet to keep straw from untwisting; you may need 2 sheets). Dab tops of strips lightly with egg wash.
Bake straws until medium golden brown, 12 to 15 minutes. Let cool 1 minute, then loosen gently from sheet with spatula. Serve warm or cool.
Note: Nutritional analysis is per straw.