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Preheat the oven to 325 degrees F. Spray a 10-inch tube pan with non-stick spray and set aside.
For the marble cake: Whisk the flour, baking powder and salt together in a small bowl and set aside.
In the bowl of a stand mixer fitted with the paddle attachment, cream the butter, oil and sugar together on medium speed until smooth, fluffy and pale yellow, 1 to 2 minutes. Add the milk and vanilla and blend together. Add the eggs, one at a time, mixing well between each addition and scraping down the sides and bottom of the mixing bowl as needed.
Reduce the mixer speed to low and add the flour mixture and blend until just combined, about 5 seconds. Pour half the batter into the tube pan, spreading it out evenly. Return the bowl to mixer, add the cocoa powder to the remaining batter, and mix until combined. Pour the chocolate batter over the batter in the pan and run a butter knife back and forth through the batter to create a marbled effect. Bake the cake until a toothpick inserted into the center comes out clean, about 1 hour and 25 minutes. Remove the cake from the oven and let it cool on a wire rack for 10 minutes, then invert it onto a large plate or cake stand and let the cake cool completely.
For the chocolate cream cheese frosting: While the cake cools, make the chocolate cream cheese frosting. In the clean bowl of a stand mixer, add the cream cheese, butter, and vanilla and beat on medium speed until smooth. Add the confectioners' sugar and the cocoa and beat on medium-low speed until the dry ingredients are incorporated, then increase the mixer speed to medium and beat until the mixture is thick and smooth. Add the milk, a tablespoon at a time, until the frosting is like a smooth, velvety buttercream. Spread the frosting generously on the top and sides of the cake with an offset spatula or butter knife. Slice and serve.