Preheat the oven to 350°F (180°C. Line two large baking sheets with parchment paper.
Add softened butter and white granulated sugar to the bowl of your stand mixer and cream them together until smooth.
Add egg and vanilla extract and mix it in.
In a separate bowl sift all purpose flour and then add baking soda, salt and matcha powder. Mix together.
Add the dry ingredients in with the wet and mix it with a spoon (not your stand mixer) into a dough.
Get in there with your hands and form it into a ball of dough. Transfer the ball of dough to a flour dusted surface.
Roll out the dough to around ¼ inch thick. Dip a cookie cutter into some flour (so that it doesn't stick) and cut out your cookies.
Place the cookies onto a parchment lined baking sheet.
Then, take the remaining dough and form it into a ball again, roll it out again and cut out the next batch of cookies. Repeat until your dough is finished. You should have 20-30 cookies (depending on the size of your cookie cutter).
Place the cookies into the oven to bake for 8-10 minutes until very lightly browned on top and on the bottoms.
Let them cool down and firm up directly on the tray.
Prepare your frosting by adding powdered sugar, butter, milk and vanilla extract to the bowl of your stand mixer and beating together until smooth.
When your cookies are completely cool, spread on the frosting generously.
Place the tray of frosted cookies into the fridge for 15 minutes for the frosting to set and your cookies are ready to serve.