Whisk together flour, baking powder, and salt in a medium bowl; set aside. In a large bowl, cream softened butter and granulated sugar until light and fluffy.

Beat in the egg and vanilla extract until well combined. Gradually add the dry ingredient mixture to the wet ingredients, mixing until just combined and a dough forms.

Divide the dough in half, flatten each half into a disc, wrap in plastic wrap, and chill for at least 1 hour or until firm.

Preheat oven to 375°F (190°C) and line baking sheets with parchment paper. On a lightly floured surface, roll out one dough disc to 1/4-inch thickness and cut shapes using cookie cutters.

Place cut cookies on prepared baking sheets and bake for 8-10 minutes, or until edges are lightly golden. Let cookies cool on the baking sheet for a few minutes before transferring to a wire rack to cool completely.

In a large bowl, beat softened butter until creamy. Gradually add powdered sugar, alternating with milk, until smooth and desired consistency is reached.

Stir in peppermint extract and a few drops of red food coloring, if using. Once cookies are completely cool, spread frosting on top of each cookie.

Immediately sprinkle generously with crushed candy canes before the frosting sets. Allow frosting to set before storing.