Separate the egg yolks and egg whites into 2 different bowls. Set aside.
In a large bowl, sift together the dry ingredients: all purpose flour, cornstarch, matcha and baking powder. Set aside.
Preheat oven to 350°F/177°C. Line a 8" by 4" loaf pan with parchment paper.
In a clean stand mixer bowl fitted with a whisk attachment, transfer the egg whites and whip until foamy. Add in the cream of tartar and keep beating until lots of bubbles form.
Gradually add in the ¼ C of granulated sugar and whip until stiff peaks. (Peak will be straight when whisk is lifted).
Remove the whipped egg whites and transfer to a large clean bowl. Set aside.
Using the same mixer bowl, change to a paddle attachment and add in the butter and sugar.
Cream until light and fluffy, about 2-3 minutes on high speed.
Gradually add in the egg yolks one by one, and then slowly add the milk and vanilla extract. Beat until combined. (At this point, it may begin to curdle, which is fine).
Remove the mixer bowl and sift about ⅓rd of the dry ingredients (flour, cornstarch, matcha, and baking powder) over the egg yolk batter.
With a spatula, fold in the dry ingredients (flour, cornstarch, matcha, baking powder) into the mixture and alternate with ⅓rd of the whipped egg whites, repeating until everything is combined together and there are no white streaks left, but making sure not to overmix as to deflate the batter.
Transfer the cake batter to the lined loaf pan. Carefully tap the pan a few times to release any trapped air bubbles.
Bake at 350°F/177°C for 60-65 mins, until golden brown on top, or until a toothpick inserted comes out clean, or with the slightest crumbs clinging to it.
Let cool completely before slicing.
Cake can be stored in an airtight container at room temperature for up to 3 days. Store in the fridge for up to 1 week.
Sliced cake can be individually wrapped in plastic wrap and frozen for up to 2 months. Just remove from the freezer about 15-20 minutes and leave at room temperature before serving.