01 - Preheat oven to 350°F. Line two baking sheets with parchment paper and set aside.
02 - In a medium bowl, whisk together the all-purpose flour, cocoa powder, baking soda, and salt until evenly distributed.
03 - In a large bowl, beat the softened butter and granulated sugar together using an electric mixer until the mixture is pale, light, and fluffy, about 2 to 3 minutes. Add the egg and vanilla extract, beating until fully incorporated.
04 - Gradually add the dry ingredient mixture to the creamed butter mixture, mixing on low speed until just combined. Avoid overmixing to keep the cookies tender.
05 - Scoop tablespoon-sized portions of dough, roll into balls, and flatten slightly between your palms. Arrange on the prepared baking sheets, spacing them about 2 inches apart. Bake for 8 to 10 minutes until the cookies are set. Allow to rest on the baking sheets for 2 minutes, then transfer to a wire cooling rack to cool completely.
06 - In a mixing bowl, beat the softened butter until smooth and creamy. Add the sifted powdered sugar, heavy cream, peppermint extract, salt, and food coloring if desired. Beat on medium speed until the filling is light, fluffy, and spreadable.
07 - Spread or pipe approximately 1 tablespoon of peppermint filling onto the flat side of half the cooled cookies. Press the remaining cookies on top, flat side down, to form even sandwiches.
08 - Refrigerate the assembled sandwich cookies for 15 minutes to allow the filling to firm up before serving.