In a large bowl, beat butter and sugar on medium speed with an electric mixer until light and fluffy. Add the egg and vanilla and beat until well combined.

In a medium bowl, whisk together the flour, cocoa powder, baking powder, and salt. Gradually add the dry ingredients to the butter mixture, mixing on the lowest speed until just combined.

Divide dough into two portions; shape each portion into a disc, wrap in plastic wrap or parchment paper and chill in the fridge for 1 hour.

Once the dough is chilled, preheat oven to 350°F. Line a baking sheet with parchment paper; set aside.

Remove one portion of the dough from the fridge; unwrap and place on a lightly floured surface. Roll to 1/8-inch thickness. Use a cookie cutter (between 1 ½ to 2 inches in diameter) to cut out cookies and place on the prepared baking sheet.

Bake cookies for 14 minutes. Transfer to a wire rack to cool completely.

In a large bowl, cream the butter with an electric mixer. Add the powdered sugar and salt, then add the milk and the extracts. Mix on medium-high speed until light and fluffy. Beat in the crushed peppermint candies until well combined.

Place filling in a piping bag fitted with a large round tip, or a zip-top bag with the corner cut out. Pipe filling onto half of the cookies, and gently top with remaining cookies to form sandwiches.