In a medium bowl, sift together the flour, espresso, and salt. Whisk together and set aside.
In the bowl of a stand mixer, add the butter and beat until creamy, about 1 minute on a medium- high speed. Add the sugar and beat until light and fluffy, about 2 more minutes. Scrape the bowl down once or twice during this process.
Scrape the bowl down and add the vanilla. Beat until combined. Add the egg and beat until thoroughly combined. With the speed on low, add the flour mixture in three parts, mixing until just combined after each.
Turn the dough out onto a large piece of cling wrap. Wrap well and form into a disk. Refrigerate for 2 hours, or overnight.
When ready to bake, preheat the oven to 325℉ and line the baking sheets with silicone baking mats or parchment paper.
Between two sheets of parchment paper, roll the dough to ¼-inch thickness. Using a 2.5 inch round cookie cutter, cut as many circles as possible and place 1 inch apart on the prepared baking sheets. Repeat with the remaining dough, gathering up the scraps, re-rolling, and cutting until all the dough is used.
Bake the cookies until they no longer look wet and are soft to the touch, 8 to 10 minutes. Remove the cookies from the oven and cool for 2 minutes before transferring them to a wire rack to cool completely.