Position racks in the upper and lower thirds of the oven; preheat to 400 degrees F. Line 2 baking sheets with parchment paper. Microwave the butter and spice mix in a small bowl until the butter melts. Let cool to room temperature.
Sprinkle 1/4 cup granulated sugar on a cutting board. Unfold the puff pastry on top of the sugar and roll out into a 10-by-12-inch rectangle. (Lightly dust the rolling pin with flour if the dough starts to stick.) Brush the spiced butter evenly on the puff pastry, then sprinkle the remaining 1/4 cup granulated sugar on top. Starting from a long side, roll the dough into a tight log. Slice the log into 18 rounds.
Place each round cut-side down. Make a slit in each round from the center of the coil to the end of the dough strip. Open up the round into a semicircle; you should have 4 stacked strips of dough in graduated sizes (the pieces will puff up and form a tree when baked). Arrange 9 semicircles about 4 inches apart on each baking sheet.
Bake, switching the pans halfway through, until the cookies are puffed and golden, 18 to 22 minutes. While still warm, carefully reshape the trees with a skewer if necessary. Let cool completely on the pans.
Make the icing: Whisk the confectioners' sugar, 1 tablespoon milk and the vanilla in a small bowl until smooth. Gradually stir in up to 1 more tablespoon milk until the icing is thick but pourable. Spoon the glaze into a small resealable plastic bag and snip off a corner. Pipe on the cookies. Decorate with coarse sugar and/or nonpareils.
Photograph by Ryan Dausch