Preheat your oven to 350°F (175°C). Line a 9x13 baking pan with parchment paper.
In a large mixer bowl or stand mixer with the whisk attachment, beat the large eggs, sugar, and vanilla extract on high speed until the mixture doubles in volume and turns pale yellow (about 5-6 minutes).
Gradually add the milk and oil into the whipped egg mixture, followed by the flour and baking powder. Use a rubber spatula to gently fold everything together.
Pour the batter into the prepared baking dish, smoothen the top with a spatula and bang the pan on the counter a couple of times to get rid of excess air bubbles.
Bake for 30-35 minutes, or until a toothpick inserted into the center comes out clean. Let the warm cake cool completely, then carefully remove it from the pan. Flip the cake so the side with parchment paper is on top. peel off the parchment paper.
Using a fork, poke holes all over the cake surface. Then in a bowl, mix the milk, heavy cream, and sweetened condensed milk. Slowly pour the mixture over the cake in batches: pour half, let it absorb, then pour the rest. Refrigerate for at least 1 hour.