1. Cake: Preheat oven to 350 degrees. Brush two 9-inch round cake pans with butter. Line bottoms of pans with parchment rounds; brush parchment with butter.

2. Whisk together flour, baking powder, and salt until thoroughly combined, 30 seconds. In a large bowl, whisk together egg whites and granulated sugar until light, opaque, and foamy, about 2 minutes. Whisk in 1 1/2 cups milk, butter, oil, and vanilla until smooth. Add flour mixture; whisk until smooth. In another large bowl, whisk together egg yolks, 1/3 cup milk, and cocoa. Add 3 1/4 cups of vanilla batter to cocoa mixture and whisk until smooth. Whisk remaining scant 3 tablespoons milk into remaining vanilla batter.

3. Spoon 1/4 cup vanilla batter into center of each prepared pan. Spoon 1/4 cup chocolate batter into each center, directly on top of vanilla batter. Repeat, always spooning batter into centers of pans, and occasionally tapping pan firmly on work surface to ensure even distribution, until all batter is used (batters should appear as concentric rings).

4. Bake, rotating pans halfway through, until cakes are puffed slightly and a tester inserted into centers comes out clean, 35 to 40 minutes. Transfer pans to a wire rack; let cool 10 minutes. Turn cakes out onto rack, remove parchment, and let cool completely.

5. Frosting: Meanwhile, whisk together cocoa, espresso powder, salt, hot water, and vanilla until a smooth paste forms. Beat butter with confectioners' sugar on medium-high speed until light and fluffy. Beat in chocolate, then cocoa mixture and corn syrup, until smooth.

6. Trim tops of cakes flat. Transfer one cake, trimmed-side up, to a cake plate or stand lined with parchment strips. Spread 3/4 cup frosting evenly over top. Place remaining cake layer on top, trimmed-side down. Spread remaining frosting evenly over top and sides of cake. Remove parchment strips; serve. Cake can be stored, uncovered, at room temperature up to 12 hours, or refrigerated, uncovered, up to 2 days; bring to room temperature before serving.