Whisk the flour, cornstarch, baking soda, and salt together in a medium bowl.

In a large bowl using a handheld mixer or stand mixer with paddle attachment, beat the butter for 1 minute on medium speed until completely smooth and creamy. Add the brown sugar and granulated sugar and mix on medium speed until fluffy and light in color. Mix in egg, vanilla extract, and peppermint extract. Scrape down the sides and bottom of the bowl as needed.

On low speed, slowly mix the dry ingredients into the wet ingredients until combined. The cookie dough will be quite thick. Add the white chocolate chips and crushed candy cane, then mix for about 5 seconds until combined. Cover dough tightly with aluminum foil or plastic wrap and chill for at least 2 hours and up to 3 days.

Remove cookie dough from the refrigerator and allow to sit at room temperature for 10 minutes, then preheat oven to 350°F (177°C). Line 2 large baking sheets with parchment paper or silicone baking mats. (Always recommended for cookies.) Set aside.

Once chilled, the dough will be slightly crumbly, but will come together if you work the dough with your hands as you roll into individual balls. Roll balls of dough, about 1.5 Tablespoons of dough each, into balls. (I like using a medium cookie scoop for this.)

Bake for 12 minutes, until lightly golden brown around the edges. Allow to cool for 5 minutes on the cookie sheet, then transfer to cooling rack to cool completely.

If using, melt the chopped white chocolate in the microwave in 20 second increments, stirring after each until completely melted. Drizzle over cooled cookies.