Preheat the Oven: Preheat your oven to 350°F (175°C). Lightly grease a 9x9 inch baking pan to prevent sticking.
Prepare the Custard: In a mixing bowl, whisk together 5 large egg yolks, 1 can (14 oz) sweetened condensed milk, 1 can (12 oz) evaporated milk, 1/2 cup whole milk, and 1 tsp vanilla extract until thoroughly combined. Pour the custard mixture evenly into the prepared pan.
Make the Cake Batter: In another bowl, sift together 1 cup all-purpose flour, 1 tsp baking powder, and 1/4 tsp salt. This ensures that there are no lumps and the ingredients are well mixed.
Beat the Egg Whites: In a large mixing bowl, beat 5 large egg whites with a hand mixer on medium speed until soft peaks form. Gradually add 3/4 cup sugar and continue beating until stiff peaks form. The mixture should be glossy and hold its shape.
Combine Wet Ingredients: In a separate bowl, mix together 5 large egg yolks, 1/4 cup vegetable oil, 1/4 cup milk, and 1 tsp vanilla extract. Once combined, gently fold this mixture in with the sifted dry ingredients and the beaten egg whites until just combined. Be careful not to overmix; you want to retain the airiness.
Layer and Bake: Carefully pour the cake batter over the custard mixture in the pan. To create a water bath, place the pan inside a larger baking dish and fill the outer dish with hot water, ensuring that it reaches halfway up the sides of the cake pan.
Bake: Place the setup in the preheated oven and bake for approximately 50-60 minutes, or until a toothpick inserted into the center comes out clean. After baking, allow the cake to cool completely in the pan before slicing.