Divide the dough in half and add chocolate and vanilla to 1 half and incorporate with hands. Add egg yolk, peppermint extract, and crushed candy to other half of dough and incorporate with hands. Cover both with plastic and chill for approximately 5 minutes. Roll out doughs separately to approximately 1/4-inch thickness. Place peppermint dough on top of chocolate and press together around the edges. Using waxed paper or flexible cutting board underneath, roll dough into log. Wrap in waxed paper and refrigerate for 2 hours.
Preheat oven to 180 degrees C.
Remove dough from the refrigerator and cut into 1/2-inch slices. Place cookies 1-inch apart on greased baking sheet, parchment, or silicone baking mat and bake for 12 to 13 minutes, rotating the pan halfway through cooking time. 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 for up to 1 week.
3 cups cake flour
3/4 teaspoon baking powder
1/4 teaspoon salt
1 cup unsalted butter, softened
1 cup sugar
1 egg, beaten
1 tablespoon milk
Icing sugar, for rolling out dough
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, wrap in waxed paper, and refrigerate for 2 hours.