In a stand mixer, beat butter until creamy. Add the matcha powder until it’s all mixed together with the butter. Don’t forget to scrape the sides.
Add the sugar and continue mixing.
While the machine is on low speed, slowly add the flour until it comes together. Gather the dough into a ball. When you are working on cutting the dough, do it in small portion at a time.
Roll out dough to 1/3-inch thickness between two pieces of parchment paper.
Cut out the cookie with a cookie cutter. With a spatula, put the cut-out cookies on parchment-lined baking sheet. Reroll by repeating step 5. Cut out the cookies until the dough is done.
Chill the dough in the refrigerator for 1 hour.
Preheat oven to 325 F. Bake for about 12 minutes or until the cookies are dry and firm to the touch. Be careful to watch the edges from being too brown.
Vanilla buttercream
In a stand mixer, beat the butter until light and fluffy.
Add the powdered sugar, vanilla extract, and salt and mix again.
Add the milk, a tablespoon at a time until you get the desired consistency.
Pipe the buttercream in between each layer of the cookies