Prepare white cake mix according to package directions. Bake in two 9-inch round pans as directed. Cool completely. You can make the white cake rounds the night before you want to assemble the cake, if you wish.
On the day you want to assemble the cake, prepare brownie mix according to package directions. Line a 9-inch round cake pan with foil and spray with cooking spray. Pour brownie batter into pan, making sure it doesn't go more than 3/4 of the way up the pan. (My brownie mix was a regular 9x13" pan size - about an 18 ounce package - and I used all of it. If the brownie mix makes more for some reason, you can bake it in cupcake liners or discard any extra.) Bake for 20-30 minutes, until a toothpick comes out with just a few crumbs about 1" from the side of the pan. Let the brownie layer cool about 30-45 minutes, then gently remove it from the pan using the foil.
Place semi-sweet chocolate and sweetened condensed milk in a small bowl. Microwave in 30 second increments, stirring between each, until smooth. Let sit 5 minutes.
Remove foil from brownies and place on cake stand. Pour chocolate sauce all over the top of the brownies (some may spill over the sides, that's okay.) Let finish cooling to room temperature.
Make the frosting by beating the butter, cream cheese, and vanilla in stand mixer fitted with the paddle attachment for about 1-2 minutes, until smooth. Add the powdered sugar, 1 cup at at time, until frosting is light and creamy. Add milk, 1 tablespoon at a time, until you reach your desired spreading consistency (I used 1 tablespoon).
Once brownie layer has cooled, place one white cake layer on top of the brownie layer. Add a thin layer of frosting on top, then top with second white cake layer. Give the entire cake a very thin layer of frosting. This is your crumb coat, so that your final coat won't be full of crumbs. Chill the cake until the frosting is set, then use remaining frosting to coat the cake. Use your spatula to make a "c" pattern over the cake for the vintage look.
Store cake loosely covered in refrigerator.