Preheat oven to 350°F/175°C. Grease two 8-inch cake pans with butter and set aside.
In a medium bowl, whisk together flour, cornstarch, matcha powder and baking powder. Sift the mixture to remove any lumps. Set aside.
In a mixing bowl fitted with whisk attachment, beat together melted butter, oil and sugar until light and fluffy, 3 to 4 minutes.
Separate egg yolks and egg whites. Add egg yolks to butter and sugar mixture bowl and beat until combined.
Slowly fold flour mixture into butter, eggs and sugar mixture until completely combined, followed by 2 tablespoons of whole milk and mix until combined.
In another mixing bowl fitted with whisk attachment, beat egg whites and cream of tartar until soft peaks form. Fold 1/3 of the egg whites into batter and fold until the mixture is just combined, then fold remaining egg whites into batter. Do not over mix.
Divide mixture between prepared pans and bake for 18-20 minutes until a toothpick inserted into the center comes out clean. Allow the cake to cool completely on a wire rack before frosting.