Put the butter, confectioners sugar, peppermint oil (or extract), and salt into a medium mixing bowl. Mix until combined using a fork or an electric mixer. Then, add the flour and mix with a rubber spatula (or electric mixer) just until a crumbly dough forms.
Squeeze the dough into small pieces and roll each piece into a 1-inch ball. If the dough seems too try to retain its shape, add a few drops of ice water and mix the dough gently with your hands to hydrate it slightly.
Place the balls on a plate, cover them with plastic wrap, and place them in the refrigerator while you preheat the oven to 350 degrees. Bake the cookies 9 to 11 minutes, until they are set and the bottoms are lightly browned.
Cool them on the cookie sheet for 1 minute before moving them to a wire rack.
Mix the crushed candy canes with ½ cup of confectioners sugar. Roll the warm cookies into the peppermint sugar and allow them to continue cooling. When they have cooled completely, roll them in the peppermint sugar a second time. Store the cookies airtight at room temperature.