In a large bowl, cream together the softened butter and granulated sugar until light and fluffy. Beat in the egg and vanilla extract until well combined.

In a separate bowl, whisk together the flour, baking powder, and salt. Gradually add the dry ingredients into the wet mixture, mixing until a dough forms.

Divide the dough into two equal portions. Leave one portion plain and add red food coloring to the other, mixing until evenly colored.

Roll out each portion of dough between two sheets of parchment paper to about ¼ inch thick. Chill the dough for 15-20 minutes.

After chilling, remove the top layer of parchment paper from each dough piece. Place the red dough over the plain dough, trim the edges, and roll the dough into a log.

Wrap the dough log in plastic wrap and chill for at least 1 hour. Preheat the oven to 350°F (175°C) and line baking sheets with parchment paper.

Slice the chilled dough into ¼-inch rounds and place them on the baking sheets. Bake for 8–10 minutes or until lightly golden. Cool completely on a wire rack.

For the filling, beat together the powdered sugar, softened butter, peppermint extract, and milk until smooth. Spread a generous amount of filling on one cookie and top with another to form a sandwich.

Optional: Roll the edges of the sandwich cookies in crushed peppermint candies for added texture and flavor.