Preheat oven to 350 degrees and grease an 8x8 baking pan.
Sift together the flour, sugar, baking powder and salt. Using an electric beater, add in the butter in small pieces, about 10 seconds apart. Beat until the butter is incorporated in small crumbs, then add the oil and continue to beat until there are no clumps and you have a fine, crumbly, cornmeal-like texture (this will take a few minutes).
Add ½ cup of the milk to the mixture and beat for 4 minutes until pale, thick and smooth (beating for this long will help create a more even consistency that will bake into a more even cake). Beat the eggs and vanilla into the remaining milk mixture and gradually add the egg mixture to the rest of the ingredients. Increase speed to medium and beat for one minute, then scrape down the sides and fold batter a few times to make sure everything is incorporated.
Divide the batter in half. Beat the matcha into half the batter until evenly incorporated.
Pour batter into prepared pan, marbling as desired, and bake for 35-40 minutes, or until edges are golden and an inserted toothpick comes out clean. Let cool before cutting into a toast shape.