On a floured surface with a floured rolling pin, roll a sheet of puff pastry out into a 16-inch square. Brush with some olive oil and sprinkle about 1 cup za'atar across the dough to mostly cover with only small gaps of dough showing, brushing off any excess, and leaving 1/2-inch border of clean dough on the top and bottom edges. Fold the dough in half towards you, pinching together the naked edges to seal. Lightly go over the dough with the rolling pin to help the dough stick together. Repeat with the remaining puff pastry.
Line 2 baking sheets with parchment paper and preheat the oven to 400 degrees F. Beat the egg white with 1 tablespoon water. Brush some over the surface of the dough and sprinkle each with an additional 1/3 cup za'atar. Using a pizza wheel or very sharp knife, slice each piece of dough crosswise into 12 strips. Gently twist each strip, then lay on the prepared baking sheet leaving at least 2 inches in between each, and refrigerate for at least 15 minutes. (Save the remaining za'atar for another use.)
Bake the chilled straws, rotating and switching positions of the pans halfway through, until golden brown, 10 to 15 minutes. Serve immediately or cool to room temperature.
If sumac is unavailable, replace with fresh lemon zest. Mix the zest with the kosher salt and allow it to dry out in a single layer overnight before combining with the other spices.