In a large mixing bowl, beat the softened butter and powdered sugar together until light and fluffy, about 2-3 minutes.
Add the egg, vanilla extract, and peppermint extract. Beat until well combined. Gradually add the flour and salt, mixing until the dough comes together.
Divide the dough into two equal portions. Leave one portion plain. Add a few drops of red food coloring to the second portion of dough and knead until the color is evenly distributed.
Wrap both portions of dough in plastic wrap and refrigerate for about 30 minutes, or until firm.
Preheat the oven to 375°F (190°C). Line baking sheets with parchment paper.
Take about a tablespoon of each dough (white and red) and roll them into 4-5 inch ropes. Gently twist the red and white dough ropes together and curve the top to form a candy cane shape. Place the shaped cookies on the prepared baking sheets.
Bake the cookies in the preheated oven for 8-10 minutes, or until the edges are just lightly golden. Be careful not to overbake, as these cookies are meant to be tender and soft. While the cookies are still warm, sprinkle them with granulated sugar for a sparkly finish.
Let the cookies cool on the baking sheets for a few minutes before transferring them to a wire rack to cool completely.