Whisk together the flour, cocoa powder, and salt in a medium bowl to combine.
Beat the butter and sugar in the bowl of an electric mixer on low speed for about 3 minutes, until the butter is the size of peas. Add the egg and mix until well combined. With the mixer on low speed, add the flour mixture in three additions, blending until just incorporated.
Turn the dough out onto a lightly floured work surface and divide in half. Shape each piece of dough into a flat disk and wrap in plastic wrap. Chill in the refrigerator for at least 1 hour, or overnight.
Preheat the oven to 325°F and line a few baking sheets with parchment paper. Transfer one disk of dough to a lightly floured work surface and roll out to 1/8-inch thickness. Using a 2-inch round cookie cutter, cut out circles of dough and arrange on the prepared baking sheets, spacing the cookies 1-inch apart. Repeat with the second half of dough. Cut a smaller circle out of the center of half of the circles of dough.
Bake for 13 to 15 minutes, until firm to the touch. Remove from the oven and let cool for 5 minutes. Transfer the cookies to a wire rack and let cool completely.
Beat the butter, powdered sugar, and peppermint extract in the bowl of an electric mixer on medium speed for 2 to 3 minutes, until smooth and creamy. If needed, add the milk, 1 tablespoon at a time, to thin out the icing. Fold in the crushed peppermints.
Spread some icing over the flat side of one cookie. Place a cookie with a hole on top and press gently to secure. Repeat with the remaining cookies and icing.