Grease and line a 9 x 13" pan with parchment paper. Preheat oven to 325 F.
In a large mixing bowl, sift together cake flour, sugar, baking soda, first set of cream of tartar, and salt. Create a well in the center, add in yolks, oil, vanilla extract, and milk. Stir to combine, then set aside.
In the bowl of a stand mixer using the whip attachment, whip the egg whites on slow speed until frothy. Then, add the second set of cream of tartar. Increase to medium speed and once whites become opaque, add sugar in a slow, steady stream. Increase to high speed, whip until stiff peaks.
Gently fold 1/3 of the meringue into the milk-flour mixture. Once incorporated, fold in the rest of the meringue.
Pour batter into prepared tin from one spot and spread the batter out evenly with an angled spatula. Tap the pan on the counter to release large air pockets.
Bake 14-16 minutes in a preheated oven until the cake springs back in the center.
Let cool 5 mins in the pan.
Place cooling rack over cake and flip cake onto the rack. Gently remove the paper. Let the cake cool a further 5 minutes. Place a fresh piece of parchment on top of cake. Flip the cake, and start rolling, layering the parchment between sections of the cake. It is important to roll the cake before it cools; otherwise it will crack .Let the cake cool wrapped, while making the chestnut cream.