In the work bowl of a stand mixer fitted with the paddle attachment - Add the butter and sugar and beat on medium speed until light an fluffy. Add the egg yolk and vanilla extract and beat until incorporated - scraping down the sides of the bowl as needed.
Turn the mixer down to low speed and sprinkle in the salt, then flour in a few increments - letting the flour mix in before adding the next increment. Scrape down the sides of the bowl as needed. After adding the last increment of flour, let the dough mix for about a minute until it starts to form a rough dough. The dough will be crumbly.
Turn off the mixer and remove half of the dough. Gently knead the dough together with your hands until it forms a ball. Set aside on a piece of plastic wrap or baking sheet.
With the remaining dough in the stand mixer bowl, add the peppermint extract and 2 drops of red food coloring. Turn the mixer on low and beat until the food coloring starts to color the dough. You can add more food coloring to get your desired color of red - I recommend only a drop at a time. Once the dough is mostly red (there can still be little streaks of white, since we are going to knead the dough into a ball), turn off the mixer and gently knead the dough together with your hands until it forms a ball.
Chilling Options - At this time, you can either wrap the two doughs separately in plastic wrap and refrigerate for at least 2 hours, or overnight. Or, you can shape the cookies (see below for shaping), place on the prepared baking sheets, cover with plastic wrap, and refrigerate for at least 2 hours, or overnight.
30 minutes before you are ready to make the cookies - remove the dough from the refrigerator and let come to room temp on the counter. Preheat the oven to 350F degrees and adjust the oven rack to the middle position. Line two baking sheets with parchment paper. Note - If you chose to refrigerate shaped cookies, you do not need the cookies to come to room temp. Skip below to the baking instructions.
Shaping the Cookies - Take 2 teaspoons of each dough at a time to make a candy cane cookie - 1. (separately) Roll each color of dough into a ball with the palms of your hands. 2. Roll each dough ball into a rope 5 inches long. 3. Gently press the two dough ropes together. 3. Roll the dough rope in one direction so it starts to twist and make the swirl. You can also hold one end of the rope and gently twist the other end to make the swirl. 4. Once you have your dough swirled, roll the rope a couple more times to smooth out any seams. 5. Gently curve the top to make the top of your candy cane. 6. Optional - sprinkle the tops of the cookies with granulated sugar or sprinkles.
Note - if you find your dough to be crumbly as you are forming the dough balls and rope, let the dough sit at room temp for a few more minutes - it may be too cold. Another option - when you are making the small dough balls, you can knead the small dough balls a little more to get the dough to a workable temperature. Don't knead too much though - we don't want the cookies to be tough.
Place your cookies 2 inches apart on the prepared baking sheets. At this point, I recommend placing your shaped cookies back in the fridge for about 10 minutes. With all of the rolling and shaping, it is good to let the dough relax and the butter to firm back up.
Baking the Cookies - One baking sheet at a time - bake the cookies for 9-11 minutes, or until the edges just start to golden. Note - while the first baking sheet of cookies are baking, refrigerate the other baking sheet of cookies until ready.
Cool the cookies on the baking sheet set on a wire cooling rack for 5 minutes, then transfer to the cooling rack to cool completely. Happy Holidays!