Sift together the flour, baking powder, and salt. Set aside.
Place the butter and sugar in the bowl of a stand mixer fitted with the paddle attacahment and beat on medium speed until light in color, about 2 minutes. Add 1 egg and the milk, and beat to combine. With the mixer on low speed, gradually add the flour, and beat until the mixture pulls away from the side of the bowl.
Divide the dough in half. To one half, add the chocolate and vanilla and incorporate with clean hands in a large mixing bowl. In a separate mixing bowl, add the egg yolk, peppermint extract, and crushed candy to other half of the dough and incorporate with clean hands. Cover each dough half with plastic wrap and chill for 10 to 15 minutes.
After the rest, roll out each half separately to approximately 1/4-inch thickness; the chocolate dough should be slightly larger. Place the peppermint dough on top of the chocolate and press together around the edges. Using parchment paper or a flexible cutting board underneath, roll the dough into a log. Wrap in parchment paper and refrigerate for 2 hours.
Heat oven to 375ºF. Line a half-sheet pan with parchment paper and set aside.
Remove dough from the refrigerator and, using a thin-bladed knife, cut into 1/2-inch slices. The dough may crack or break during cutting; use your fingers to work the dough back together and continue slicing. Place cookies 1 inch apart on the prepared pan and bake for 12 to 13 minutes, rotating the pan halfway through cooking time. Remove from oven and leave the pan for 2 minutes, then move to a wire rack to cool completely. Store in an airtight container for up to 1 week.