Preheat oven to 350°F. In a medium bowl, mix flour and ground coffee to combine. Set aside.
In the bowl of a standing electric mixer, cream together butter and sugar with the paddle attachment. Add egg, vanilla, and salt and mix to combine. Add flour mixture in several additions, stirring on slow speed until just combined. Press clumps of dough together, kneading gently to bring the dough into a cohesive mass. It's okay if it feels a bit dry and crumbly.
Divide the dough in half. Roll each piece between two sheets of parchment paper to a thickness of about 3/8-inch (removing and replacing the paper on both sides of the dough periodically makes rolling easier). Remove the top piece of parchment and cut out cookies with a 2-inch star cutter, avoiding the edges if they are dry and crumbly. Place cookies on parchment-lined baking sheets and bake until the edges just begin to brown, about 10 to 12 minutes.
Press the dough scraps together and re-roll between parchment paper, stamping stars, and repeating with remaining dough.
Cool completely before decorating.
Whisk powdered sugar, milk, and vanilla extract together in a large bowl. The icing should be thick -- if you lift the whisk, the ribbon of icing that falls back into the bowl should hold its shape for a few seconds before recombining with the mixture. Adjust with additional confectioners' sugar or milk to achieve desired consistency.
Working one at a time, pipe or spread a thin layer of icing on cooled cookies. Set iced cookies on a rack over a rimmed, parchment-lined baking pan. Sprinkle generously with sprinkles. Repeat with one sprinkle color until you're ready to switch. Use the parchment paper to return excess sprinkles to the container to reduce waste.
Let cookies sit for about six hours or overnight to allow the icing and sprinkles to firm up before serving.