To make the matcha cake, sift the flour, cornstarch, and salt into a medium bowl, then set aside.
Put the eggs and sugar in the bowl of a stand mixer, then place the bowl over a medium saucepan of simmering water, making sure that the bottom of the bowl is not touching the water. Whisk continuously by hand for 1 to 2 minutes, until the sugar has dissolved and the mixture is warm to touch.
Attach the bowl to the stand mixer fitted with the whisk attachment and whisk on high speed for 7 to 8 minutes, until the mixture has tripled in volume and is cool to touch. Decrease the speed to low and whisk for 2 minutes more to remove any large air bubbles—the mixture should look silky and smooth at the end.
Meanwhile, heat the butter, vegetable oil, and milk in a small saucepan over low heat. When the butter has melted, remove from the heat and add the matcha powder. Whisk until smooth, then set aside.
When the egg mixture has tripled in volume, scrape it into a large wide bowl—this will make it easier to fold the batter later. Scoop about 1/2 cup (120 ml) of the egg mix into the pan with the matcha milk and fold until combined—you don’t need to be too careful to retain the air here—then fold this back into the large bowl with the remaining egg mixture. Now sift half the (pre-sifted) flour mix over the top, and carefully fold it through the egg mix, taking care not to lose too much air. Sift the remaining flour over and fold gently until incorporated.
Pour the batter into the lined cake pan and tap the pan firmly on the work surface a couple of times to remove any large air bubbles. Bake for about 30 minutes, or until the center of the cake feels springy to the touch. It should look uniformly light brown and well risen. Remove from the oven and immediately drop the pan twice on a heatproof work surface from about an 8-inch 20 cm) height—this feels counterintuitive, I know, but it helps to prevent the cake from sinking in the middle. Place the cake on a wire rack to cool completely in the pan.