1. Preheat the oven to 350°F and line two baking sheets with parchment paper.

2. In a large bowl, beat 1 cup softened butter and granulated sugar until light and fluffy.

3. Mix in the egg and 2 teaspoons vanilla extract until smooth.

4. In a separate bowl, whisk together the flour, baking powder, and salt.

5. Add the dry ingredients to the wet ingredients and mix until a soft dough forms.

6. Scoop the dough into 2 tablespoon portions, roll into balls, and place on the prepared baking sheets.

7. Gently flatten each dough ball slightly.

8. Bake for 9 to 11 minutes, until the edges are set but the tops are still pale.

9. Cool on the baking sheet for 5 minutes, then transfer to a wire rack and cool completely.

10. For the frosting, beat 1/2 cup softened butter until creamy.

11. Add powdered sugar, milk, 1 teaspoon vanilla extract, and peppermint extract, then beat until fluffy and smooth.

12. Mix in the pink food coloring until the frosting is evenly tinted.

13. Spread the frosting over each cooled cookie.

14. Sprinkle crushed candy canes over the tops.

15. Let the frosting set before serving or storing.