Preheat the oven to 350 F. Line the bottom of an 8-inch square baking pan with parchment paper.
In a medium bowl, whisk together cake flour, baking powder, and salt until combined.
In a large mixing bowl, whisk together egg yolks, ¼ cup of the sugar, oil, water, and vanilla extract until pale and frothy.
Sift half of the flour mixture over the yolk batter and stir well using a whisk. Sift the remaining half and beat the mixture well until it is smooth and lump-free.
Meanwhile, beat the egg whites and cream of tartar in the stand mixer. Once soft peaks form, gradually add the remaining ¼ cup of sugar while the mixer is still running until stiff peaks form.
Fold the beaten egg whites gradually into the yolk batter using a rubber spatula. The mixture should be uniform and color.
Pour the batter into the prepared pan. Tap the pan lightly against the counter to get rid of any air pockets.
Bake the cake at 350 F for 15-18 minutes or until the cake springs back to the touch.