Preheat the oven to 400ºF (200ºC) (if you have a convection oven, turn it on).
Line two baking sheets with parchment paper or silicone baking mats. Set aside
With a handheld or stand mixer with the paddle attached, add cubed butter, brown sugar and granulated sugar.
Turn mixer on low for a few seconds, then gradually increase speed to medium-high, mixing for about 3-4 minutes until well incorporated and fluffy.
Turn mixer to low and add egg and egg yolk then vanilla and peppermint extracts.
Mix on medium for about 30 seconds until well incorporated, stopping to scrape down the sides and bottom with a rubber spatula, as needed.
With the mixer off, add in both flours, salt, baking soda and baking powder.
Turn mixer on low and stir until just combined. Do not over mix.
Add in white and semisweet chocolate chips and crushed candy canes, stirring to combine.
Turn mixer off and knead dough a bit with clean hands to ensure mix-ins are evenly incorporated.
Portion dough into 2 ounce size balls, rolling between your palms to ensure a smooth ball.
Place about 2 inches apart on prepared baking sheets.
Bake in a preheated oven for 4-5 minutes if using convection (7-8 minutes in conventional oven) then rotate baking trays and bake for another 4-5 minutes for convection (7-8 minutes for conventional).
Cookies will be barely set in the middle and brown around the edges.
Allow to cool for a 15 minutes on trays (cookies will continue to bake a bit on the trays) then transfer to wire racks to cool completely.