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In the bowl of an electric mixer fitted with the paddle attachment, beat the butter until smooth. Add in the granulated sugar and mix until creamed.
With the mixer on low, add in the dry ingredients in two batches. Mix until just combined and the dough comes together.
Split the dough in half and place each piece on a sheet of plastic wrap. Roll or press the dough into 1/2-inch-thick disks and refrigerate until firm, about 2 hours.
Once the dough has been chilled, preheat the oven to 350 degrees F.
Working with one piece of dough at a time, roll out to about 1/8-inch thick between two pieces of parchment paper. Dust with a good amount of flour to prevent sticking.
Use a fluted 3-inch round biscuit or cookie cutter to cut out cookie rounds. Carefully transfer to an ungreased nonstick cookie sheet with an offset metal spatula. (If the dough becomes sticky or soft and hard to work with during rolling and cutting out, then transfer to the fridge or freezer for a few minutes to chill and firm up.)
Continue with the rest of the dough, re-balling then chilling the scraps in between.
Use a star or other small cutter to cut shapes out of half of the cookies. These will become the tops of the sandwiches. Save and bake the stars, or ball up with the dough scraps.
Bake until the dough is set and no longer shiny, 10 to 12 minutes. Let cool on the baking sheets for a few minutes, and then transfer to a wire rack to cool completely.
Meanwhile, for the filling: Combine the hot water and espresso powder. Stir until dissolved. Using an electric mixer fitted with a paddle attachment, beat the butter until smooth. With the mixer on low, add the confectioners' sugar, espresso mixture and vanilla. Beat together until smooth.
To assemble: Spread about 2 slightly heaping teaspoons of filling on the bottom halves of each cookie with a small offset spatula, knife or back of a spoon. Top with the other half of the cookies (with the shapes cut out). Press together gently. If the filling is a bit soft, chill in an airtight container until set, about 20 minutes. Store in an airtight container in a cool, dry place.