For the cake: Preheat the oven to 350 degrees F. Butter the bottoms and sides of two 9-inch round cake pans; dust with flour, and tap out the excess.
Whisk together the flour, cornstarch, baking powder and salt in a medium bowl; set aside.
Beat the butter and sugar in a large bowl with an electric mixer on medium-high speed until light and fluffy, about 5 minutes. Beat in the eggs, one at a time, making sure the batter is well mixed before adding the next egg. Beat in the vanilla until blended.
Alternate folding the flour mixture and the milk into the butter mixture with a spatula, adding the flour in three additions and the milk in two, starting and ending with the flour, until just smooth. (The batter may look slightly curdled as you mix it, but this is not a problem.) Divide the batter evenly between the prepared pans.
Bake until the cakes are golden and the centers spring back to the touch, 35 to 40 minutes. Let cool in the pans about 10 minutes, turn out onto cooling racks and let cool completely. Turn the cakes over, and poke the tops all over with the tines of a fork. Generously brush the tops with the liqueur, letting it soak in before brushing on more.
For the frosting: While the cakes cool, bring a few inches of water to a simmer in a medium saucepan. Set a large heatproof bowl over it (do not let the bowl touch the water), add the egg whites and sugar and whisk until the mixture is warm and the sugar has dissolved, 3 to 4 minutes. Remove the bowl from the pan, and beat the mixture with an electric mixer on medium-high speed until stiff, glossy peaks form, 4 to 7 minutes. Beat in the butter a few pieces at a time, and continue beating until the frosting is smooth, 1 to 2 minutes. (Don't worry if the mixture becomes soupy or looks separated; keep beating and it will come together.) Stir the Kahlua and instant coffee together until dissolved, and beat into the frosting with the mixer running on low speed. Increase the speed to medium-high, and beat until fluffy, 1 to 2 minutes.
To assemble: Place 1 cake layer right-side up on a cake plate or stand, and spread with 1 cup of the frosting. Top with the second cake layer. Frost the top and sides of the cake with the remaining frosting.
If your kitchen is hot and your frosting becomes too soft and thin for spreading, refrigerate it for 20 to 30 minutes and then beat it until smooth.
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