Preheat oven to 350 degrees F. Line two baking sheets with parchment paper.
Combine flour with baking powder, baking soda, and salt in a medium bowl. Whisk together buttermilk, yogurt, and vanilla in another bowl.
Beat together softened butter with granulated sugar in the bowl of a stand mixer until light and fluffy, about 3 minutes. Add 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.
Use an ice cream scooper or a ¼ cup measuring cup and scoop the batter onto the prepared baking sheet. Leave at least 4 inches in between the cookies, giving them room to spread, about 5 cookies per baking sheet. Chill the cookies in the refridgerator 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. Divide the powdered sugar into two bowls. Add 2 tablespoons of coconut milk to one bowl for the coconut frosting. Mix until well combined. The icing will be thick but spreadable.
Whisk together remaining 3 tablespoons coconut milk with matcha powder until matcha is completely dissolved. Add the matcha milk to the other half of the powdered sugar and mix until well combined.
Spread the coconut 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.
Notes
These matcha black and white cookies are slightly thicker than traditional black and white cookies. If you want your cookies to spread more, skip the chilling time.