1. In a large bowl, combine the flour, baking soda, salt, and spices; set aside. In the bowl of a stand mixer, beat the butter and sugars together until they appear light and fluffy. Add the egg and mix to combine, scraping down the sides of the bowl once if needed.
2. Working in three batches, add the dry ingredients, mixing between each addition until just incorporated. Remove the dough and turn it out onto a sheet of plastic wrap; form into a dish, wrap tightly and refrigerate until firm, at least 2 hours. Dough can be made up to 1 day ahead.
3. Preheat the oven to 350˚. Remove the dough and cut it in half. Rewrap one half and place back in the fridge; let the other sit at room temperature for a few minutes to soften slightly. Lightly flour a work surface and your rolling pin and roll out the dough to ¼ inch thick.
4. Line a baking sheet with parchment paper. Using cookie cutters, cut out the gingerbread men, and transfer to the prepared sheets. If the dough feels very soft, refrigerate the gingerbread men on the baking sheet for 15 minutes before placing in the oven to bake. Bake until cookies appear firm and like they’re just starting to crisp, 12-15 minutes. Remove and let cool on the sheet 5 minutes, then transfer to racks to cool completely.
5. Meanwhile make the sandwich filling. Beat together the butter, vanilla, and salt until fluffy. Add the confectioners’ sugar in 2 batches and beat on low speed until incorporated. Add the milk and beat until the mixture is thick but spreadable, adding more milk in1 teaspoon increments as needed. Spread some of the icing onto half of the cooled cookies, sandwich each with another cookie.