In the bowl of a stand mixer fitted with a paddle attachment, cream together the butter, granulated sugar, powdered sugar, and salt until thoroughly combined and light and fluffy, about 3 minutes.
1 cup unsalted butter, ¼ cup granulated sugar, ¼ cup powdered sugar, ¼ teaspoon salt
In a separate bowl, whisk together the all-purpose and cake flours. With the mixer on low, add the flour mixture into the butter mixture in 3 additions, scraping down the sides of the bowl in between each addition
1 ¼ cups all-purpose flour, 1 cup cake flour
Tear off a large piece of plastic wrap and place it on a clean countertop. Dump the shortbread dough out onto the plastic wrap and form the dough into a round with your hands (doesn’t have to be perfect). Wrap the dough tightly in plastic wrap and refrigerate for at least 4 hours or overnight
Preheat the oven to 350ºF and line 2 cookie sheets with parchment paper or silicone baking mats. Set aside
Lightly flour a clean work surface. Unwrap cookie dough and place on the floured surface. With a lightly floured rolling pin, roll out the dough to about ½-inch thickness. If the dough is too solid to roll, whack it a few times with the rolling pin to loosen the dough a bit. Or simply wait 5-10 minutes to allow the cookie dough to warm a bit. Using a star-shaped cookie cutter, cut the refrigerated dough (dip the cookie cutter in flour if the dough is sticking to it) and place on baking sheet 1 inch apart (cookies will not spread much). Bake in the preheated oven until the cookies are just starting to turn golden at the edges, about 15-16 minutes. Remove from the oven and allow cool on the baking sheets
In two small microwavable bowls, melt separately the white chocolate and dark chocolate in 30 second intervals on 50% power until melted and smooth. Using a fork, dip the tines into the chocolate and drizzle over top of the cookies. Immediately sprinkle with crushed candy canes before the chocolate sets
white chocolate baking bar* (see note) and dark chocolate baking bar or chips, crushed candy canes