Preheat oven to 350 degrees F. Line two baking sheets with parchment paper.
Combine 1 ⅓ cups flour with ½ teaspoon baking powder, ¼ teaspoon baking soda, and ½ teaspoon salt in a medium bowl. Combine ⅓ cup buttermilk with 2 tablespoon yogurt and 1 teaspoon vanilla in another bowl.
Cream together ½ cup softened butter with ½ cup granulated sugar until light and fluffy, about 3 minutes. Add 1 egg and beat until well combined.
Add the wet ingredients, and mix until well combined. Add the dry ingredients, stirring just until combined. The batter should be fairly thick.
Ice an ice cream scooper or greased ¼ cup measuring cup and scoop the batter onto the prepared baking sheet. Leave at least 2 inches in between the cookies, giving them room to spread. Chill the cookies in the fridge for 30 minutes.
Bake the cookies for 15-17 minutes or until lightly golden on top and golden brown on the bottom. Let the cookies rest for 5 minutes on the baking tray. Flip the cookies over and transfer to a wire rack to cool completely.
Meanwhile, make the icing. Beat together 2 ¼ cups powdered sugar with ¼ cup milk until well combined. The icing will be thick but spreadable. Reserve half of the frosting and add 1 tablespoon matcha and 1 tablespoon milk, stirring until well combined.
Spread the vanilla frosting on one half of each of the cookies. Let sit for 10 minutes, then spread the matcha frosting on the other half. Let sit for 10 minutes. Serve.