Combine soft butter, powdered sugar, vanilla extract, peppermint extract and salt in a large bowl if using a handheld mixer (or mixing by hand with a wide rubber spatula) or in the bowl of a stand mixer fitted with a paddle attachment and mix for 2 minutes on medium speed until very smooth, pale and creamy. Scrape down the sides and bottom of the bowl.
Add milk and mix until well incorporated and the mixture looks smooth, silky and glossy. Add flour and mix on low speed until it starts to absorb and a dough starts to form. First it will look a bit dry and crumbly and then slowly it will come together. Before it is completely incorporated, add crushed candy cane and mini white chocolate chips or finely chopped white chocolate and mix it in until the dough is smooth.
Cover the bowl tightly and chill in the fridge for 30 minutes while you preheat the oven.
Place the powdered sugar for rolling into a shallow bowl.
Scoop up heaped tablespoons of cookie dough and roll into smooth balls with your hands. You should get 13-15 cookies. Place dough balls onto prepared baking sheets, spacing them at least 2 inches apart. Bake for about 15 minutes until golden brown on the bottom edges and just barely browned on top. They wont get too much color on top.
Transfer baking sheets to a wire rack and let cookies to cool for 15-20 minutes on the trays, then very gently roll them in the powdered sugar to coat completely. The powdered sugar will melt a bit and get sticky on the outside of the cookies. Place the cookies on wire racks to cool completely, then roll in powdered sugar again so they are nicely coated.
Cookies stay fresh covered at room temperature for up to 1 week.