Cocoa Peppermint Sandwich Cookies

Note the prep time includes a minimum of 1 hour of inactive chill time for the dough.

For the cookies:
Whisk together the flour, cocoa powder and salt in a medium bowl.

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. You can re-roll the scraps of dough to get additional cookies. 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 cookies cool on the pan for 5 minutes. Transfer the cookies to a wire rack and let them cool completely.

For the icing:
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 of the solid cookies. Place a cookie with a hole on top and press gently to secure. Repeat with the remaining cookies and icing.