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Preheat the oven to 350 degrees. Spray 2 (8-inch) round cake pans with cooking spray. Be sure your pans are at least 2 inches tall, because the cake layers rise quite a bit. Line the bottom of the pans with parchment paper, then spray the paper with more cooking spray. Set pans aside.
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Combine the cake mix, flour, sugar, sour cream, water, vegetable oil, egg whites, and vanilla extract in a large mixing bowl.
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Beat with an electric mixer on medium speed for 2 minutes—batter should be very smooth. Use a rubber spatula to scrape up from the bottom and get everything totally mixed together.
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Divide cake batter between the prepared pans and bake cake layers 43 to 48 minutes, until a cake tester inserted into the center of the cakes comes out clean.
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Cool cake layers for 20 minutes in pans, then run a knife around the edge of the pans and carefully invert onto a wire rack to cool completely before assembling your cake.