Portion the dry ingredients (flour, baking powder, salt) in a bowl, set aside.
Portion the wet ingredients (egg, vanilla and milk) in another bowl, set aside.
Cream the butter and sugar in your mixer on a medium speed. Add the wet ingredients to the butter mixture in three stages. Mix until fully incorporated for each stage. When the last of the wet is incorporated, turn the mixer off.
Add the dry ingredients all at once. Mix on speed 1 just until the dough starts coming together. Give it a 5 second flash on a medium-high speed, to insure the mixture is homogeneous.
Divide the dough in half.
Melt chocolate and butter at 50% power in microwave in 1 minute increments and stir to ensure its fully melted. Allow to cool a bit, then mix the chocolate and vanilla in with half of the dough with hands. Form a square and wrap chocolate dough in plastic wrap.
Add egg yolk, peppermint extract, and crushed candy canes to the other half of the dough and mix with hands. Form a square and wrap peppermint dough (this will be much sticker) in plastic wrap.
Chill both doughs for 2 hours to overnight.
Roll out chocolate dough on confectioners sugared surface to a long rectangle shape, approximately 1/4-inch thick. Place peppermint dough on top of chocolate and spread using fingers to completely and evenly cover the chocolate dough. Press together around the edges. Using hands, roll dough into log, working your way from one end and then back again until completely rolled. Wrap in plastic wrap and refrigerate for 2 hours.
Preheat oven to 375 degrees F.
Remove dough from the refrigerator and cut into 1/2-inch slices. Place cookies 1 inch apart on greased, parchment lined, or silicone baking mat lined baking sheet and bake for 12 to 13 minutes, rotating the pan halfway through cooking time. Continue to chill remaining dough until ready to cut and bake. Remove from oven and let sit on baking sheet for 2 minutes, then move to a wire rack to cool completely. Store in an airtight container and/or freeze.