Pre-heat oven to 350 degrees.
Combine ¾ cup sugar and butter in a mixer (or large bowl with hand mixer) and cream together at medium speed until light and fluff, 2-3 minutes.
Add egg, vanilla, and mint extract and blend to combine.
In a separate medium bowl, whisk together flour, cream of tartar, baking soda, and salt.
Slowly combine the dry ingredients to the butter/sugar mixture and blend on low speed, until just combined, taking care not to over-mix.
Fold in ¼ cup of the crushed candy canes.
In a small bowl, mix the remaining ¼ cup of crushed candy cane with the remaining ¼ cup sugar.
Shape the dough into ¾ inch balls and roll each in the sugar/candy mixture. Place the dough balls about 2 inches apart on a parchment paper-lined baking sheet.
Bake cookies for 8-10 minutes, until edges are just starting to turn golden. Let cookies cool on baking sheet for a couple minutes before moving them to a wire rack to cool completely.