In a medium-sized mixing bowl, whisk together the flour, cocoa powder, and salt. In a stand mixer fitted with a paddle attachment, beat the sugar and softened butter on medium speed for approximately 3 minutes, until the mixture is light and fluffy. Add the egg and mix until fully incorporated. Gradually add the dry ingredients, mixing on low speed just until combined. Form the dough into a disk, wrap in plastic wrap, and refrigerate for at least 1 hour.

Preheat the oven to 350°F (175°C). Line two baking sheets with parchment paper. Scoop heaping teaspoons of dough and roll them into smooth balls. Arrange them on the prepared baking sheets, spaced about 1 inch apart. Gently flatten each dough ball using the bottom of a drinking glass. Bake for 8 minutes, or until the surface appears set and a slight indentation remains when lightly touched. Allow cookies to cool on the baking sheet for 3 minutes before transferring to a wire rack to cool completely.

In a clean stand mixer bowl fitted with the paddle attachment, combine the powdered sugar and softened butter. Mix on low speed until the sugar is incorporated, then increase to medium speed and beat until light and fluffy. Add peppermint extract and food coloring, mixing until the desired color and consistency are achieved.

Spread or pipe a generous amount of peppermint filling onto the flat side of one cookie. Top with a second cookie, flat side down, and press gently to sandwich them together. Repeat with remaining cookies.

Place the crushed peppermint candies in a shallow bowl. Roll the edges of each cookie sandwich in the candy pieces, gently pressing to ensure they adhere.