Prep: Preheat your oven to 400 degrees F. Line two baking sheets with parchment paper, or silicon mats, and set them aside.
Dough: In a large bowl, cream the butter and sugar using an electric mixer, for 1 minute. Add in the egg, milk, and vanilla extract. Mix again until smooth.
In a separate bowl combine the dry ingredients, the flour, baking soda, and salt. Add half of the dry ingredients to the bowl of wet ingredients, mix. Continue to add the other half of the dry ingredients and the crushed peppermint candy. Mix again until well combined and no white streaks of flour show. The dough will be sticker than normal cookie dough.
Scoop the cookie dough using a 2 TBSP sized scoop, placing them onto the parchment-lined cookie sheets. Bake: Place the cookies in the preheated oven and bake for 9-10 minutes. Let them cool for an additional two minutes on the cookie sheets before transferring them to a cooling rack.
Peppermint Vanilla Buttercream: In a mixing bowl, whip the butter on medium speed for about 3 minutes, until whipped and light in color.
Add in the vanilla and peppermint extract, milk, and 1 cup of powdered sugar. Mix for 1 minute, then add in the remaining 2 cups of powdered sugar and mix on medium speed for 3-4 minutes. Add additional milk, if needed, until the desired consistency is achieved.
Once the cookies are fully cool, frost each one with about 1 TBSP of the homemade buttercream frosting. Then top with more crushed candy cane pieces. Serve immediately and enjoy!