Mix the sugar, sesame seeds and cinnamon in a small bowl. Lay out a 12-by-16-inch piece of parchment paper. Sprinkle 1/2 cup of the sugar mixture evenly on the paper. Lay the puff pastry on top and sprinkle with another 1/2 cup of the sugar mixture. Roll out the dough into a 12-inch square, pressing the sugar into the dough with the rolling pin. Roll up the dough tightly into a log. Wrap in the parchment paper and freeze until firm, about 30 minutes.
Preheat the oven to 350 degrees F. Line 2 baking sheets with parchment paper. Slice the log into 1/2-inch-thick rounds and place 1 inch apart on the prepared baking sheets. (If the spirals unravel, pinch them back together.) Freeze until firm, about 20 minutes.
Bake until the pastry is puffed and light brown, 10 to 15 minutes. Remove from the oven and sprinkle with the remaining sugar mixture. Return to the oven and bake until the cookies are glazed and golden, 10 to 15 more minutes. Use a spatula to transfer the warm cookies to another sheet of parchment paper, flipping them over before the sugar hardens. Let cool completely. Store in an airtight container up to 1 week.