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In another bowl, stir together the flour, ground coffee, baking powder and salt. On low speed, slowly add the dry ingredients to the butter mixture and mix until completely incorporated and a soft dough has formed.
Place a rectangular sheet of parchment or wax paper on a clean surface. Pat the dough out into a rectangular shape on top of it. Cover with another sheet of parchment and roll the dough to 1/4- to 1/2-inch thickness. Remove the top sheet of parchment and transfer the dough to a baking sheet. Refrigerate for a minimum of 1 hour or overnight.
Preheat the oven to 350 degrees F. Line two baking sheets with parchment paper or silicone baking mats.
Cut the dough into rounds (or desired shape) using cookie or biscuit cutters. Transfer the cutouts to the baking sheets, spacing them about 2 inches apart. Gather the scraps, roll again and cut out more rounds (refrigerate to chill the dough if it gets too soft).
Bake for 10 to 15 minutes, depending on thickness and shape. Do not let the cookies brown; start checking for doneness early. The finished cookies will have slightly golden brown bottoms, but they will feel very soft and under-baked until they cool. Hold the cookies on the baking sheets for 10 minutes, and then transfer to a wire rack to cool completely before frosting.
For the Irish cream Italian meringue buttercream: In a small, heavy saucepan, stir together 1 cup of the sugar with 1/4 cup cold water to combine. Heat the mixture on medium-high heat. Dip a pastry brush in water and brush down the sides of the pan to remove any excess sugar, and then partially cover with a lid to prevent crystallization. Cook the sugar mixture until the temperature registers 245 degrees F on a candy thermometer.
While the sugar mixture is heating, make the meringue. Place the egg whites in the bowl of a stand mixer fitted with the whisk attachment. Beat the egg whites on high speed until medium peaks form. Slowly add the remaining 1/4 cup sugar and continue beating until stiff peaks form, about 5 minutes total.
With the mixer running, slowly pour the hot syrup into the meringue. Continue beating at high speed for 2 minutes. Reduce the speed to medium and beat until the meringue has cooled to room temperature and the bowl is cool to the touch, about 15 minutes.
Begin adding the butter two to three cubes at a time. Increase the mixer speed to high; add the Irish cream liqueur and coffee extract and beat until the butter and liquids are fully incorporated. (The buttercream can be held in the refrigerator for up to one week or frozen for one month.)