In a large bowl, whisk together the cake flour, bread flour, baking powder, baking soda, and kosher salt. Set aside.
To the bowl of a stand mixer fitted with the paddle attachment, add the butter, brown sugar, and granulated sugar. Cream on medium speed for 3-5 minutes, or until the mixture is light and fluffy.
Reduce the mixer speed to low. Add the eggs one at a time, mixing until each is fully incorporated. Mix in the vanilla extract.
Gradually add the dry ingredients to the wet mixture, mixing on low speed until just combined. Avoid overmixing.
Mix in the chocolate chips and crushed peppermint candies until evenly distributed.
Cover the dough and chill it in the refrigerator overnight.
When ready, remove the cookie dough from the refrigerator and let it come to room temperature.
Preheat oven to 350°F (175°C). Line 6 baking sheets with parchment paper.
Using a 2-tablespoon scoop, drop dough onto the lined baking sheets, spacing them about 2 inches apart. Sprinkle additional crushed peppermint candy on top of each.
Bake for 10-12 minutes, or until the edges are golden and the centers are just set.
Let the cookies cool on the baking sheet for 5 minutes, then transfer them to a wire rack to cool completely.