Let the butter come to room temperature. Crush the candy canes in a zip-top bag with a rolling pin, and keep any water away from them.
For the dough
Whisk the flour, baking powder and salt in a bowl. In a second bowl, whisk the eggs, milk and vanilla extract.
Cream the butter and sugar in a stand mixer on medium speed until light and fluffy. Add the egg mixture in three additions, mixing well after each one. Add the flour mixture all at once and mix on low just until the dough starts coming together, then give it a 5-second burst on medium-high. Divide the dough in half.
For the chocolate half
Melt the chocolate and butter in the microwave at 50% power in 1-minute increments, stirring until smooth. Let it cool until just warm, then work it and the vanilla into one half of the dough with your hands. Shape it into a flat square and wrap it in plastic wrap.
For the peppermint half
Work the egg yolk, peppermint extract and crushed candy canes into the other half of the dough with your hands. Shape it into a flat square and wrap it. Chill both doughs for at least 2 hours or up to overnight.
To roll
Divide each chilled dough in half. On a counter dusted with powdered sugar, roll one piece of chocolate dough into a rectangle about 12 inches long, 5 inches wide and 1/4 inch thick. Set half of the peppermint dough on top and spread it with your fingers to cover the chocolate completely and evenly. Press the edges together.
Roll it up tightly, starting from a long side, into a log about 12 inches long. Wrap it in plastic wrap and repeat with the rest of the dough to make a second log. Chill both logs for 2 hours.
To bake
Preheat the oven to 375°F and line baking sheets with parchment or silicone mats. Cut each log into 1/2-inch slices and place them 1 inch apart. Bake for 12 to 13 minutes, turning the pan halfway through. Keep the dough you aren't slicing in the fridge. Let the cookies sit on the pan for 2 minutes, then move them to a wire rack to cool completely.
Notes
Adapted from Alton Brown's Chocolate Peppermint Pinwheel Cookies.
Store baked cookies in an airtight container at room temperature for up to 1 week, away from other cookies so they don't all taste like peppermint. Baked cookies can be frozen for up to 3 months.
Make ahead: wrap the unbaked logs tightly and refrigerate for up to 3 days or freeze for up to 3 months.