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Preheat the oven to 350 degrees. Mix the wet ingredients in a medium mixing bowl. Mix in the flour. Pour the batter into a greased rectangular pan. Bake the cake. Let the cake cool for 10 minutes. Invert the cake onto a tray.
Put the butter, sugar, vanilla, eggs, and milk in a bowl. Beat with a whisk or an electric mixer until the mixture is smooth and creamy. Slowly add the flour to the wet ingredient mixture, stirring it in well with a whisk or electric mixer. Keep beating the mixture until you no longer see specks of white flour. Use butter or cooking spray to oil a 13" x 9" pan and pour the batter into it. Put the pan in the oven and bake the cake until a toothpick inserted in the center comes out clean, about 20 minutes.  Allow it to finish cooling completely while you make the icing.