Put a large piece of parchment paper on a work surface and sprinkle with 2 tablespoons of the vanilla sugar. Lay the puff pastry on top, sprinkle with 2 more tablespoons vanilla sugar and roll out into a 12-inch square. Trim the edges to make an even square, then cut the pastry into four 3-inch-wide strips.
Lightly beat the egg and water in a small bowl. Brush the egg wash lengthwise down the center of 3 of the pastry strips. Stack the 3 strips neatly on top of each other; top with the remaining pastry strip. Place the rolling pin lengthwise down the center of the pastry stack and press gently to adhere the layers in the center; the edges should remain separated. Cut the stack crosswise into 1/2-by-3-inch strips. (If the pastry is too soft to cut, transfer to a baking sheet and refrigerate until firm, about 15 minutes.)
Line 2 baking sheets with parchment paper. Pinch the center of each strip, then twist the pastry twice at the pinch point. Arrange 1 inch apart on the prepared baking sheets. Fan out the edges of the pastry. Lightly brush with the egg wash, then sprinkle with the remaining 2 tablespoons vanilla sugar. Refrigerate until firm, about 20 minutes.
Position racks in the middle and lower thirds of the oven; preheat to 375 degrees F. Bake, switching the pans halfway through, until the cookies are golden brown, 18 to 20 minutes. Immediately transfer the cookies to racks to cool completely.