In the bowl of a stand mixer, cream together butter and sugar. Add egg, salt and peppermint extract; beat well until incorporated.
Gently mix in flour. Chill cookie dough for at least one hour and up to overnight. Mix together powdered sugar and crushed candy canes; set aside.
Preheat oven to 375 degrees F. Form dough into one-inch balls and place on parchment or Silpat lined baking sheets. Using a one-inch cookie scoop, form dough into balls. Bake for 10 to 12 minutes until bottoms start to brown.
Immediately roll warm cookies in powdered sugar mixture. Cool completely and store covered. Makes 3 to 4 dozen.