Make cake: Preheat oven to 350°F. Grease and flour 2 9-inch round cake pans that are at least 2 inches deep.
In a large bowl, using an electric mixer on medium-low speed, mix flour, sugar, cocoa, baking soda, baking powder and salt until well combined. In a medium bowl, whisk eggs, sour cream, milk, oil and vanilla together until smooth. Mix egg mixture into flour mixture on low speed, scraping down sides of bowl once or twice, until smooth. Dissolve espresso powder in 1/2 cup of very hot water and stir into batter. Divide batter evenly between pans. Bake, rotating pans halfway through, until a toothpick inserted in center of a cake comes out clean, 40 to 45 minutes.
Let cakes cool in pans on a wire rack for 30 minutes. Run a paring knife around inside edges of pan and turn cakes out onto rack to cool completely.
Make frosting: Put egg whites, sugar, cream of tartar and salt in a large, clean metal bowl set over a saucepan filled with 1 inch of simmering water (do not let water boil or allow bottom of bowl to touch water). Adjust heat as needed to maintain a simmer. Cook mixture over medium-low heat, beating with an electric mixer on medium-high speed, until frosting holds stiff peaks, about 7 minutes. Remove bowl from heat, add vanilla and continue to beat until frosting is glossy and billowy, 1 to 2 minutes longer.
Assemble cake: Level cake layers with a serrated knife, if desired (see instructions below). Place 1 cake layer, bottom side up, on a serving dish. Spread evenly with about 2 cups frosting. Top with second cake layer, bottom side up. Spread remaining frosting evenly over top and sides of cake.
Frost Like a Pro
Follow these simple steps and people might think you bought your dessert at a bakery.
Step 1. Level it.
If your cake has a domed top, use a serrated knife to cut off the dome, making top as level as possible. Flip the bottom layer over so it rests on the cut side before frosting.
Step 2. Frost the middle.
Spread the frosting evenly on the first layer using an offset spatula. Don't worry about making it look perfect, as long as it's even. Be careful not to pull crumbs into the frosting. Place the second layer on top and gently press down.
Step 3. Finish it.
Spread the frosting over the top and sides. To make pretty whorls, run the back of a tablespoon over the frosting in a swirling motion.