Whisk together flour, baking soda, and salt in a medium bowl; set aside.

In a large bowl, cream softened butter and granulated sugar until light and fluffy, then beat in the egg and vanilla extract.

Gradually add the dry ingredients to the wet ingredients, mixing until just combined to form the cookie dough.

Chill the dough for at least 30 minutes, then roll it out on a floured surface and cut into desired shapes.

Bake at 375°F (190°C) for 8-10 minutes, or until the edges are lightly golden; transfer to a wire rack and cool completely.

Beat softened butter in a medium bowl until creamy, then gradually add powdered sugar, milk, vanilla extract, and peppermint extract until the frosting is smooth.

Stir in ½ cup of crushed candy canes until evenly distributed.

Once the cookies are completely cool, spread an even layer of candy cane frosting over each.

Garnish with additional crushed candy canes for a festive touch.