Preheat the oven to 350 degrees. Grease two 9-inch round cake pans heavily with butter, line the bottoms with parchment paper, and dust the sides with cocoa powder.

In a large bowl, whisk together the flour, granulated sugar, cocoa powder, baking powder, baking soda, and salt.

In a separate bowl, whisk together the milk, vegetable oil, egg yolks, vanilla extract, and the precious Black Walnut flavoring. Pour the wet ingredients into the dry ingredients and beat with a mixer on medium speed for exactly two minutes.

Turn the mixer to its lowest speed and carefully stream in the hot, freshly brewed coffee. The batter will become very thin; this is correct.

In a pristine glass bowl, use an electric hand mixer to whip the reserved egg whites until stiff peaks form.

Using a rubber spatula, ruthlessly, yet gently, fold the stiff egg whites into the thin chocolate batter until no white streaks remain. This is the century-old secret to the cake's structural lift.

Divide the batter evenly between the prepared cake pans. Bake for 25 to 30 minutes, or until a wooden skewer inserted into the center emerges with moist crumbs, not wet batter.

Cool the cakes in the pans for 10 minutes, then turn them out onto a wire rack to cool completely.

In a medium bowl, pour the melted butter over the cocoa powder and stir until a thick paste forms. Alternately add the confectioners' sugar and milk, beating until the frosting is incredibly smooth and spreadable. Stir in the vanilla and Black Walnut flavorings.

Frost the cooled cake generously, swirling the frosting with the back of a spoon to create dramatic, swooping peaks.