Preheat the oven to 350°F. Lightly grease an 18” X 13” jelly roll pan with cooking spray and line it with a non-stick baking mat (the same size as the pan) or a fitted piece of parchment paper. Set aside.
Place the eggs and sugar in the bowl of an electric mixer fitted with a whisk attachment. Beat the egg mixture on medium-high speed until the batter reaches ribbon stage, 8 to 10 minutes. It is done when the mixture has tripled in volume, is very pale, and thick. When you lift the whisk from the batter and move it around, the batter should hold two figure-eights before melting back into the bowl.
Stop the mixer and sift the flour, cornstarch, baking powder, and salt straight into the bowl. Use the whisk attachment to fold the dry ingredients into the batter, making sure to incorporate any dry pockets at the bottom of the bowl.
In a separate bowl, combine the oil, vanilla extract, and about 1 cup of the batter. Stir together until thoroughly combined. Add the oil mixture into the batter and fold together with a flexible rubber spatula until combined.
Tip the batter into the prepared pan and gently spread it out into an even layer with an offset spatula. Bake for 12 to 14 minutes, rotating halfway if your oven heats inconsistently. When done, the surface of the sponge cake should start to brown and spring back when lightly poked with a finger.
While the cake bakes, set up your rolling station. Gather a clean dish towel, fine-mesh sieve, large cutting board or cooling rack (about the same size or larger than the baking sheet), and a thin paring knife or metal spatula. You will need to roll up the cake before it cools or the cake may crack.
When the cake is done, remove it from the oven. Sift confectioners’ sugar all over the top of the cake. Run the knife or metal spatula around the edges of the cake to release it from the pan. Place the clean towel on top, followed by the cutting board. Carefully flip everything upside-down (using oven mitts, if the pan is still hot to the touch). The cake should release onto the towel. Remove the pan and set aside.
Remove the parchment paper or baking mat from the bottom of the cake. Sift confectioners’ sugar all over the bottom of the cake. Starting with one of the long edges, gently roll the cake up lengthwise within the towel. You should have a long log of cake with the towel rolled into the spiral. Allow the cake to cool inside of the towel, seam-side down. This will take about an hour.
Meanwhile, make the coffee soak.