Preheat oven to 325 degrees F (165 degrees C). Grease an 8-inch baking pan.
Whisk cocoa powder, flour, espresso powder, and salt together in a bowl.
Combine egg yolks, white sugar, and water in the bowl of a stand mixer. Beat on high speed until eggs are light and creamy, about 2 minutes. Reduce speed to low and add butter and vanilla extract. Increase speed to medium; beat until light and fluffy, about 1 minute. Reduce speed to low and add the cocoa mixture in 3 batches, mixing thoroughly after each addition and scraping down the sides of the bowl. Stir milk slowly into the batter until well combined. Pour chocolate batter into a separate bowl.
Clean and dry out bowl and beaters. Beat egg whites and cream of tartar together on high speed until stiff peaks form. Fold 3/4 of the egg white mixture into the chocolate batter using a spatula; pour back into the egg white bowl, folding mixture in slowly until batter is smooth but still light and fluffy. Pour batter into the prepared pan.
Bake in the preheated oven until edges are set and center still jiggles slightly, about 50 minutes. Let cake cool completely, at least 30 minutes.