If green tea isn't finely ground, grind with a mortar and pestle.
Whisk dry ingredients (flour, cornstarch, matcha and green tea) together and set aside.
Cream butter and sugars with paddle attachment, on medium-low speed, until fluffy and pale in color (about 5 minutes).
Add egg, extract, and vanilla bean paste, and mix on low speed until thoroughly combined. Scrape down the paddle with a spatula, and then continue to mix on low speed for another 30 seconds to ensure everything is fully incorporated.
Add dry ingredients, all at once, and mix on low speed just until dough starts to come together and no dry bits remain. Do not overmix.
Turn dough out onto cling wrap, tightly wrap, and refrigerate overnight (or for at least 4 hours).
Knead, roll, cut out cookies, and return to fridge or freezer until firm. Leave a couple inches between cookies.
Bake at 350°F for 15 minutes, rotate pan, and bake for another 1-5 minutes until edges are starting to turn golden brown.
Remove from oven and allow to fully cool on cookie sheet before handling.