Arrange racks on the upper and lower thirds of the oven and preheat to 400 degrees F.

Stir together the granulated sugar and cinnamon in a small bowl. Beat the egg with a pinch of salt in another small bowl. Set aside.

Lightly dust a work surface with flour and roll one sheet of the puff pastry into an 18-by-10-inch rectangle with a short side closest to you. Brush the pastry lightly with the egg, sprinkle a heaping ¼ cup of the sugar mixture on top and fold in the long ends in thirds like a letter. Gently roll the pastry to lengthen it into a 22-by-4-inch rectangle. Use a pizza wheel to cut the pastry into four long strips.

Place a baking sheet-size piece of parchment paper in front of you with a long side closest to you. On the left side of the parchment, stand up one strip of dough on its side so it is cut-side up and you see the thin strip of cinnamon-sugar filling, and form the strip into a loose oval spiral, keeping about a ¼-inch space between each layer of the spiral. Use a second strip of dough to continue to wrap and form the same loose spiral.

Create a second loose oval spiral with the remaining 2 long strips on the right side of the parchment, about 4 inches apart from the first spiral.

Use a rolling pin to lightly roll each spiral into a flattened 8-by-4-inch oval. Transfer the parchment to a baking sheet.

Place a second baking sheet-size piece of parchment paper in front of you. Repeat to make 2 more flattened ovals using the second piece of puff pastry, egg wash and a heaping ¼ cup of the sugar mixture. Transfer the parchment to a second baking sheet.

Brush the 4 flattened ovals with the egg wash and sprinkle with the remaining cinnamon sugar. Bake, rotating the pans from top to bottom halfway through, until the pastry is puffed, crisp and golden, 22 to 26 minutes. Let the pastries cool on the baking sheets for 5 minutes, then remove to a wire rack to cool completely, about 1 hour.

Once cooled, whisk together the confectioners’ sugar, clear vanilla extract, a pinch of salt and 3 tablespoons of the milk in a medium bowl until smooth. The glaze should be thick but able to drizzle; if it cannot be drizzled add up to 1 tablespoon more of milk little by little to thin it out. Drizzle the glaze over the top of the pastries. Let set, about 30 minutes.