Preheat the oven to 355F (180C). Line an 11 x 16-inch (28 x 40 cm) jelly roll pan with parchment paper. Use nonstick silicone-coated parchment, a Teflon mat, or a perforated silicone mat. Alternatively, fold parchment paper to create a pan of the required size with sides about 1 inch (2 cm) tall, then place it on a baking sheet.
In a large bowl, combine the egg yolks, milk, oil, and vanilla extract. Whisk until well mixed.
Sift in the flour, matcha powder, and salt. Whisk until smooth and no dry ingredients remain.
In the bowl of a stand mixer fitted with a whisk attachment, or using a hand mixer, add the egg whites and cream of tartar. Whip on medium-low speed, gradually adding the sugar. Once all the sugar has been added, increase the speed to medium-high and whip to medium peaks. Do not overwhip to stiff peaks.
Using a rubber spatula, fold one-third of the egg white mixture into the egg yolk mixture until combined.
Add the remaining egg whites and gently fold until the batter is homogeneous.
Pour the batter into the prepared pan and spread it evenly using an offset spatula. Bake for 12 to 15 minutes, or until the top just begins to turn lightly golden.
Remove from the oven and let rest for 1 minute. Turn the cake out onto a fresh sheet of parchment paper and carefully peel off the parchment from the bottom.
Roll the cake up with the parchment paper, rolling loosely to prevent cracking. Leave it rolled and allow it to cool completely to room temperature.