Preheat the oven to 350 degrees. Butter a 9×13-inch pan, lining the bottom with a piece of parchment paper cut to fit the pan.
Pour the egg whites into the bowl of your mixer and beat on medium-high speed for 4 to 5 minutes, or until they hold soft peaks. Slowly stir in the sugar and continue beating until they hold harder or stiffer peaks. Turn off the mixer and, with a spatula, move the egg white mixture into a large mixing bowl.
Rinse the mixer bowl and its whisk. Now, pour the egg yolks into the bowl and beat on medium-high speed for about 5 to 6 minutes, or until the yolks become creamy and puffy and their color has toned down to an almost cream color rather than a loud yellow. Stir in the vanilla and continue beating for another minute.
Pour the egg yolk mixture onto the egg white mixture and, with a spatula, in circular motions, combine them into a homogeneous single batter. Do so gently, trying not to lose too much volume gained from beaten egg whites. Fold in the flour, scraping the bowl with the spatula so that all the flour is well mixed.
Pour batter into the prepared cake pan and place into the oven for 22 to 25 minutes, or until a toothpick comes out clean. It can be a bit moist, but not wet. The top of the cake should be tanned or golden brown.
Once it cools down, turn it onto a platter. Remove parchment paper, cover the top with an upside-down platter and invert again. The platter should be large enough to hold the cake and the vanilla sauce you are about to prepare. Using a fork, or two, poke holes all over the cake so that it will better absorb the vanilla sauce.
In a mixing bowl, combine the sweetened condensed milk, evaporated milk, milk, and vanilla extract. Pour the vanilla sauce over the cake. It may appear like too much sauce, but it will all be absorbed!
In the bowl of your mixer, whip up the heavy cream with the confectioners’ sugar on medium-high speed until the mixture holds up stiff peaks. Spread the whipped cream all over the already-wet cake and place it in the refrigerator. You can decorate the cake with berries or any other topping of your choice.