Preheat oven to 325 degrees and line two baking sheets with parchment paper or Silpats.
In the bowl of a stand mixer or in a large bowl with a hand mixer, cream together butter, brown sugar, and ½ cup granulated sugar until light and fluffy, 2-3 minutes.
Add sour cream, egg, and peppermint extract and beat to combine.
In a separate large bowl, whisk together flour, cocoa powder, baking powder, baking soda, and salt.
Pour dry ingredients into wet mixture and beat until just combined, taking care not to over-mix.
Place remaining ¼ cup sugar in a small bowl and place confectioners' sugar in a separate small bowl. Using about 2 tablespoons of dough at a time, roll dough into balls. Roll each ball into granulated sugar, followed by confectioners' sugar. Place on prepared baking sheets about 2 inches apart.
Bake cookies for about 16 minutes, until set. They should still be a little soft. Let cookies cool in pan for a couple minutes before moving to rack to cool completely.
Once cookies are completely cooled, drizzle with sour cream icing and sprinkle with crushed candy canes.