For the cake: Preheat the oven to 350 degrees, with the rack in the middle. Line the bottom of a 10-inch tube pan with parchment and coat the pan and paper with nonstick spray.
Cut the butter and shortening into thick slices and add to an electric mixer fitted with the paddle attachment. Cream for 3 or 4 minutes on medium speed, then add the sugars, 1/2 cup at a time, beating 1 to 2 minutes between each addition. Add the eggs one at a time, and beat 1 to 2 minutes between each addition.
Whisk the flour and baking powder together. With the mixer on low speed, add the flour mixture and milk, alternately, to the batter, in 3 additions. Raise speed and mix for an additional minute. Lower speed again, add the vanilla, and beat another minute.
Scrape batter into the prepared tube pan. Bake for 70 minutes, or until a skewer inserted close to the center withdraws clean. Cool in the pan for 15 minutes, then invert onto a wire rack. Cool completely before frosting.
For frosting: Wash the mixing bowl and paddle, then re-attach to the mixer. Cream the butter and cream cheese on medium speed until light and creamy, then add the vanilla. Add the powdered sugar 1/2 cup at a time, beating until smooth after each addition. When cake is completely cool, frost.