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Heat the oven to 350 degrees F and arrange the rack in the middle.
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Coat 2 (8-inch) cake pans with butter and flour, and tap out any excess flour.
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Set aside.
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Combine flour, baking powder, salt, and baking soda in a small bowl and whisk to aerate and break up any lumps.
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Set aside.
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Place butter in the bowl of a stand mixer fitted with a paddle attachment and mix on medium-high speed until fluffy and light in color, about 3 minutes.
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Add sugar and vanilla, and continue to beat another 5 minutes.
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Add yolks one at a time, letting each incorporate fully before adding the next.
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Stop the mixer and scrape down the sides of the bowl and the paddle.
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Return the mixer to medium-high speed and add whole eggs in the same manner, allowing 1 minute between additions.
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Stop the mixer again and scrape down the sides of the bowl and the paddle.
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(The mixture should be very pale in color.)
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Add 1/3 of the flour mixture, and turn the mixer to low speed, mixing until the flour is just incorporated.
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Add 1/2 of the milk, and mix until just incorporated.
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Continue with remaining flour mixture and milk, alternating between each, until all ingredients are incorporated and smooth.
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Divide batter evenly between the prepared pans.
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Bake until cake edges slightly pull away from the pans and a cake tester inserted in the center comes out dry with just a few crumbs, about 35 to 40 minutes.
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Remove the pans from the oven and let cool on a wire rack, about 10 to 15 minutes.
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Run a knife around the perimeter of each cake, and turn out onto the rack to cool completely (at least 1 1/2 hours) before frosting.
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To frost, place a cake layer on an 8-inch cardboard round, a tart-pan bottom, or a cake plate.
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Evenly spread about 1/3 of the frosting over the top of the layer.
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Stack the second layer, and evenly spread another 1/3 of the frosting over the top and sides of the whole cake.
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(Dont worry about looks at this pointthis is just a basecoat, a.k.a.
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a crumb layer, and it will be covered up later.)
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Place in the refrigerator until frosting is set up and slightly hard, about 15 minutes.
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Remove from the refrigerator and spread remaining frosting over the top and sides of the cake, ensuring its as even as possible.
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Serve.