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

2. In a medium bowl, whisk together the flour, baking powder, and salt.

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

4. Mix in the egg and vanilla extract until smooth.

5. Gradually add the dry ingredients and mix just until a soft dough forms.

6. Scoop the dough into 1 1/2 tablespoon portions, roll into balls, and gently flatten them slightly on the prepared baking sheets.

7. Bake for 9 to 11 minutes, until the edges are set but not browned.

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

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

10. Add the powdered sugar, heavy cream, peppermint extract, and pink food coloring, then beat until fluffy and smooth.

11. Spread the frosting over the cooled cookies.

12. Sprinkle crushed candy canes on top and let the frosting set before serving.