In a medium bowl, sift together the flour, cornstarch, baking soda, and sea salt. Set this aside for now.
In the bowl of a stand mixer, beat the butter and sugar together on medium-high speed until light and fluffy. Then add the egg, vanilla, and peppermint extract and continue beating on medium speed until smooth and creamy. Turn the mixer off and add the flour mixture. Gently fold the flour in until it is just combined, being careful to not over-mix the dough. And then fold in the white chocolate and crushed candy canes.
Cover and chill the dough for at least half an hour before baking. Once you're ready to bake, preheat the oven to 350°F and line two cookie sheets with parchment paper.
Scoop 1½ tbsp sized balls of dough onto the cookie sheets, making sure to space the cookies at least two inches apart. Bake for 11-12 minutes, or until the edges are golden. Remove them from the oven and let them rest for 5 minutes on the cookie sheet before transferring to a wire rack to cool completely.
Optional: make a white chocolate drizzle to go on top by melting ¼ cup of white chocolate in the microwave. Melt it in 20 second intervals, stirring between each until the chocolate is smooth.