Mix the dry ingredients in a medium bowl and set aside.

Cream the softened butter and granulated sugar together using an electric mixer for 3-5 minutes until light and fluffy.

Add the egg, vanilla extract, and peppermint extract to the butter mixture and mix until well combined.

Gradually add the dry ingredients to the wet mixture and stir gently until just combined.

Wrap the dough in plastic wrap and chill in the refrigerator for 30 minutes.

Preheat the oven to 350°F (175°C). Roll out the chilled dough to about ¼ inch thick and cut out desired shapes.

Place the cut-out cookies on prepared baking sheets, spacing them about 2 inches apart.

Bake for 10-12 minutes until edges are lightly golden and centers remain soft. Cool on baking sheets for 5 minutes before transferring to a wire rack.

Whisk together powdered sugar and milk until smooth to make the frosting.

Frost the cooled cookies and sprinkle with crushed peppermint candies or candy canes.