In a stand mixer bowl, add in the butter and sugar and cream until light and fluffy, about 2-3 minutes.
Add in the beaten egg, vanilla and peppermint extract and beat until everything is incorporated.
Sift in the matcha and flour and mix until just combined. Be careful not to overmix.
Dump the mixture onto a large piece of parchment paper and form into a disc. Wrap up the disc tightly with the parchment paper and place into the fridge for at least 1 hour.
Preheat oven to 350°F.
Using a rolling pin, roll out the disc to ¼" inch thickness and cut out shapes using a round cookie cutter.
Place the cut-out cookies onto a baking sheet lined with parchment paper. Bake the cookies at 350°F for 10-12 minutes.
Remove and cool on the baking sheet for 5 minutes before transferring to a cooling rack to cool completely before decorating.
Melt the white chocolate in a microwave safe bowl for about 20-30 seconds. Use a spoon to drizzle the white chocolate over the cookies and decorate with crushed candy cane.
Cookies can be stored in an airtight container for up to 1 week.