Preheat your oven to 350°F (175°C) and lightly grease a 9×13-inch baking dish. Set it aside while you prepare the batter.
In a medium-sized mixing bowl, sift together the all-purpose flour, baking powder, and kosher salt. Give it a quick whisk to evenly combine the dry ingredients and set the bowl aside.
In the bowl of a stand mixer fitted with the paddle attachment, beat the egg yolks and granulated sugar on medium-high speed until the mixture becomes pale and thick, nearly doubling in volume. Once it reaches this airy texture, add in the vanilla extract and whole milk, mixing on low just until everything is smoothly incorporated. Gently fold this yolk mixture into the prepared dry ingredients, being careful not to overmix.
Thoroughly wash and dry the mixing bowl before whipping the egg whites—any trace of yolk or moisture will hinder their rise. Switch to the whisk attachment and whip the egg whites on high speed until stiff peaks form, which should take about 3 to 4 minutes.
Working in two additions, gently fold the whipped egg whites into the batter. Take care not to deflate the mixture; use sweeping, gentle motions until no streaks of white remain.
Pour the finished batter into your greased baking pan, smoothing out the surface evenly. Bake in the center rack for 18 to 20 minutes, or until the top is lightly golden and a toothpick inserted comes out clean. Let the cake cool in the pan for about 30 minutes.