Beat 6 egg whites in a large bowl with an electric mixer until fluffy; set yolks aside. Gradually beat in sugar until mixture holds soft peaks. Beat in flour gradually, then beat in milk, reserved egg yolks, melted butter, 4 teaspoons vanilla extract, and baking powder, beating in each ingredient thoroughly before adding the next. Pour batter into prepared baking pan.
Bake in preheated oven until cake is browned and crusty on top, no more than 30 minutes. Let cake cool in pan for about 10 minutes before gently removing cake to finish cooling.
Whisk sweetened condensed milk, evaporated milk, and half-and-half in a bowl. Pour half the 3-milk mixture into a pan deeper and longer than the cake. Lay the cake into the milk mixture. Use a fork to score deep lines into the top of the cake and pour remaining 3-milk mixture over cake, allowing the mixture to soak in.
Beat cream cheese with 1 teaspoon vanilla extract in a bowl with electric mixer until soft and fluffy; gradually beat in cream until frosting is thick and spreadable. Spread over the cake.
