Line an 8 in x 8in (20cm x 20cm) cake pan with aluminum foil. Sieve bread flour and matcha together twice so that matcha is evenly mixed into the flour. Preheat oven to 325°F (165°C).
Place egg whites in mixing bowl and beat at high speed (speed 10*) for 30 seconds until slightly foamy. Add sugar in 2 to 3 increments until firm peaks form, about 4 minutes.
Add yolks one at a time at medium low speed (speed 4*) until well combined, about 1 minute.
Add sifted bread flour and matcha mixture and beat until just combined. Add mirin and continue to beat for another minute.
Pour batter through a sieve into cake pan. Press with a spatula to help batter go through sieve. Tap pan on the counter to remove air bubbles. Using a spatula, smoothen the top of batter to remove any remaining bubbles.
Bake in preheated oven for 50 minutes or until top is evenly browned. Toothpick inserted into cake should come out clean.
Remove pan from oven and drop it from a one foot height onto the counter to prevent shrinkage. Place a piece of parchment paper on a large piece of shrink wrap. When cake is cool enough to handle, lift cake up by holding the edge of aluminum foil. Turn cake upside down onto parchment paper. Peel off aluminum foil. Wrap shrink wrap over cake and leave overnight in the refrigerator (right side up) to preserve moisture in the cake and for flavor to develop. Alternatively, warm cake can also be put into a Ziploc bag with the foil and left in the refrigerator (right side up) overnight.
To serve, cut off edges on all sides except the top and bottom with a very sharp knife. Then cut into thick slices using a sawing motion.