Preheat your oven to 350°F. Lightly grease the bottom and sides of a rimmed baking sheet (10x15-inch) and line it with parchment paper.
In a medium bowl, whisk flour, cocoa, espresso powder, and baking powder.
In the bowl of an electric stand mixer fitted with the whisk attachment, beat eggs on medium-high speed for 3 minutes until frothy and pale.
Gradually add sugar and beat until tripled in volume and fluffy, about 4-5 more minutes. When you lift the whisk out of the mixture, it should slowly fall back into itself in thick, billowy ribbons. Mix in vanilla, salt, and oil.
Sift in the dry ingredients then use a rubber spatula to fold the ingredients until just combined. Make sure to scoop the bottom of the bowl until there are no more streaks of flour. Pour the batter into the prepared pan and spread it into a very thin layer. Since the cake is very thin, it only takes 10-11 minutes to bake through. Let it cool for 10 minutes.
Run a knife along the edges of the pan to release the cake. Invert it onto a clean surface or parchment paper, then gently peel off the parchment and slice it into 3 equal rectangles. Transfer to the fridge to cool it even further.