Position racks in the upper and lower thirds of the oven; preheat to 350 degrees F. Line 2 baking sheets with parchment paper. Coarsely grind the coffee beans in a coffee grinder. Combine 1 tablespoon ground coffee in a small bowl with the turbinado sugar; set aside. Transfer the remaining coffee to a medium bowl and whisk in the flour, salt, baking powder and cinnamon.
Beat the butter and brown sugar in a large bowl with a mixer on medium-high speed until light and fluffy, about 3 minutes. Beat in the egg. Reduce the mixer speed to low and beat in the flour mixture until combined.
Roll tablespoonfuls of dough into balls, then roll each ball in the coffee-sugar mixture. Arrange 2 inches apart on the prepared pans.
Bake, switching the pans halfway through, until the cookies are just set around the edges but the centers are still soft, about 18 minutes. Let cool 3 minutes on the pans, then transfer to racks to cool completely.
Make the glaze: Whisk the confectioners' sugar and 2 teaspoons milk in a small bowl until smooth. Gradually stir in up to 2 more teaspoons milk until the glaze is thick but pourable. Spoon the glaze over the cookies. Let set, about 10 minutes.