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.
Brush pastry all over with olive oil. Sprinkle left half of pastry liberally with half of the zaatar and fold right (unsprinkled) pastry half over zaatar, like closing a book. Lightly sprinkle top of pastry with flour and roll back out into a rectangle almost as large as the one you began with. As before, brush pastry with olive oil, sprinkle half with remaining zaatar, and fold unsprinkled half over zaatar. 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, with Garlicky Yogurt Dip.
Note: Nutritional analysis is per straw.