Place an oven rack in the center position and preheat the oven to 350 degrees. Line a cupcake tin with cupcake liners.
In a medium bowl whisk together the cake flour, baking powder, baking soda, salt and matcha. Set aside.
In another medium bowl, whisk together butter, sugar, yogurt, milk, egg yolks and vanilla extract.
Whisk the dry mix into the wet ingredients and mix just until combined and no lumps remain.
Using a stand mixer fitted with the whisk attachment (or hand-held mixer), beat the egg whites until they form medium peaks. Fold the egg white mixture into the batter until fully incorporated.
Pour batter evenly into prepared cupcake pan, and bake for about 15 minutes, until a toothpick comes out crumb-free.
Remove from oven and let cool completely on a wire rack before frosting.
For the frosting:
Mix powdered sugar, matcha and butter on low speed in a stand mixer or use a hand-held eletric mixer in medium bowl until blended. Add vanilla and 1 tablespoon half-and-half. Gradually add just enough remaining half-and-half to make frosting smooth. Add a few drops of half-and-half if frosting is too thick. If it's too thin, beat in a small amount of powdered sugar.