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1
Bring the apple cider, cinnamon stick and allspice to a simmer in a small pot over medium-high heat.
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2
Cook until reduced to 1/4 cup, about 12 minutes; set aside to cool.
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3
Remove the cinnamon stick and allspice berries.
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4
Whisk the flour, ground cinnamon, baking powder and baking soda in a medium bowl.
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5
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.
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6
Add the egg and half of the cider mixture and beat until combined.
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7
Reduce the mixer speed to low; add the flour mixture and beat until just incorporated.
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8
Cover with plastic wrap and refrigerate until firm, about 1 hour.
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9
Position racks in the upper and lower thirds of the oven and preheat to 350 degrees F. Line 2 baking sheets with parchment paper.
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10
Form the dough into 3/4-inch balls and arrange 2 inches apart on the prepared baking sheets.
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11
Bake, switching the pans halfway through, until golden brown, 9 to 11 minutes.
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12
Let cool 5 minutes on the baking sheets, then transfer to racks to cool completely.
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13
Make the glaze: Whisk the confectioners' sugar and the remaining cider mixture in a small bowl until smooth.
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14
Spread the glaze on the cookies and top with the coarse sugar.
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15
Let set 30 minutes.
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16
Photograph by Andrew Purcell