Line 2 medium sheet pans with parchment paper and make some space to put them in the fridge.
In a large bowl, add the 1/2 cup salted butter, 3/4 cup granulated sugar, 2 teaspoons vanilla extract, 1 teaspoon peppermint extract, and 1 large egg. Beat with an electric mixer until light and fluffy.
In a medium bowl, sift the 1 2/3 cup all-purpose flour and 1½ teaspoon baking soda.
Add the sifted flour to the butter-egg mixture and using a spatula, stir until just combined.
Fold in the 1/2 cup chopped candy canes. Mix again to incorporate.
Scoop the dough, rolling into same-sized balls, into your prepared sheet pans; keep about 2 inches of space between the balls.
Refrigerate for 30 minutes.
Preheat the oven to 350 F. Bake the cookies directly from the fridge for 9-11 minutes or until the center is still soft. Cool before serving.