Sift together flour, baking powder, and salt. Set aside. Place butter and sugar in large bowl of electric stand mixer and beat until light in color. Add egg and milk and beat to combine. Put mixer on low speed, gradually add flour, and beat until mixture pulls away from the side of the bowl.
Divide the dough in half.
By hand mix the melted chocolate and vanilla into one half and the peppermint extract, egg yolk and candy cane bits into the other half. Make sure the candy can pieces are very small (I recommend using a food processor to crush them) otherwise your cookies will have pits where the candy melted down.
Wrap dough in saran wrap or wax paper and chill for 30 minutes.
Roll the doughs out separately to approximately 1/4 inch thickness (1/4 inch thick rubber bands that snap onto your rolling pin are great for this). Make the chocolate layer just a little longer than the peppermint.
Place the chocolate layer on wax paper and the peppermint layer on top of the chocolate. Press the edges together and then carefully roll them into a log using the wax paper as a stabilizer for the process. Refrigerate the log for 2 hours so it will hold its shape when cut into rounds.
Preheat oven to 375 degrees F.
Cut the log into equally sized rounds and place on either parchment paper or a silicone baking mat. Bake 7 to 9 minutes or until cookies are just beginning to turn brown around the edges. The let them sit on the baking sheet for 2 minutes after removal from oven before moving to a wire rack to finish cooling.