Bring the apple cider, cinnamon stick and allspice to a simmer in a small pot over medium-high heat. Cook until reduced to 1/4 cup, about 12 minutes; set aside to cool. Remove the cinnamon stick and allspice berries.
Whisk the flour, ground cinnamon, baking powder and baking soda in a medium bowl. Beat the butter and brown sugar in a large bowl with a mixer on medium-high speed until light and fluffy, 3 to 5 minutes. Add the egg and half of the cider mixture and beat until combined. Reduce the mixer speed to low; add the flour mixture and beat until just incorporated. Cover with plastic wrap and refrigerate until firm, about 1 hour.
Position racks in the upper and lower thirds of the oven and preheat to 350 degrees F. Line 2 baking sheets with parchment paper. Form the dough into 3/4-inch balls and arrange 2 inches apart on the prepared baking sheets. Bake, switching the pans halfway through, until golden brown, 9 to 11 minutes. Let cool 5 minutes on the baking sheets, then transfer to racks to cool completely.
Make the glaze: Whisk the confectioners' sugar and the remaining cider mixture in a small bowl until smooth. Spread the glaze on the cookies and top with the coarse sugar. Let set 30 minutes.