In a medium bowl, whisk the flour, cornstarch, baking soda, and salt together.
Using a standing mixer beat the butter for 1 minute on medium speed until completely smooth and creamy.
Add both sugars. Mix on medium speed until fluffy and light in color.
Next mix in the egg, vanilla, and peppermint extract.
Scrape the sides and bottom of the bowl as needed.
Slowly mix the dry ingredients into the wet ingredients on slow speed until combined.
The cookie dough will be very thick.
Add the 3/4 C white chocolate chips, 1/2 C chopped peppermint bark white chocolate chips and 1/2 C crushed candy cane, mixing for about 5 seconds until combined.
Cover dough tightly with plastic wrap.
Chill for at least 2 hours.
Remove cookie dough from the refrigerator and allow to sit at room temperature for 10 minutes.
Preheat oven to 350 degrees.
Line 2 large cookie sheets with parchment paper.
Set aside.
Using a small ice cream scoop, scoop out some dough and roll into a ball
Place 12 cookies onto the baking sheet and bake for 10-12 minutes or until slightly golden brown
Allow to cool for 5 minutes on the cookie sheet.
Transfer to a wire rack to cool completely.
Add the 1/2 C white chocolate chip into a bowl.
Put the bowl in the microwave melting the white chocolate in 20-second increments.
Stir after each 20- second increment until completely melted.
Drizzle melted white chocolate over cooled cookies and sprinkle with remaining crushed candy canes.