Preheat the oven to 325 degrees F.

In a bowl, beat together the evaporated milk, condensed milk, vanilla extract, egg whites, and whole eggs. Set aside. In a medium saute pan on medium heat, heat the sugar until syrup-like and brown in color, about 8 to 10 minutes. Keep stirring the sugar as it can burn easily. Pour the melted sugar caramel into a 9 by 13-inch baking dish, coating around the bottom and sides. Pour in the milk mixture. Place the entire baking dish in a larger dish filled halfway up the smaller baking dish with water, but be sure not to let any water spill into the flan mixture. Cover with foil. Bake in the oven for 45 minutes to 1 hour, or until the custard is set. Cook's Note: Insert a knife into the center, and if it comes out clean, it is fully cooked.

Let the custard cool 2 hours before serving.