Cream together butter and sugar for 1 minute. Add in egg, milk and vanilla and mix again until smooth.
Combine dry ingredients and add 1/2 to the mixing bowl, mix, then add the other half. Add in peppermint candy and mix to combine. Dough will be softer & stickier than normal cookie dough.
Scoop out about 2 TBSP of cookie dough onto a parchment lined cookie sheet. Bake for 9-10 minutes and let cool for an additional 2 minutes on the cookie sheet. Transfer to a cooling rack.
Once fully cool, frost with about 1 TBSP of peppermint vanilla buttercream per cookie. Top with crushed candy cane pieces. Store cookies in an airtight container.
*For peppermint candy I used a combination of peppermint baking chips and crushed candy cane pieces.
Peppermint Vanilla Buttercream
Mix butter in a mixing bowl on medium power for about 3 minutes, until whipped and light in color.
Add in vanilla, peppermint extract, milk and 1 cup of the powdered sugar.
Mix for 1 minute. Add in the other 2 cups of powdered sugar and mix on medium for 3-4 minutes, adding additional milk if needed until desired consistency is achieved.