Step 1
Preheat oven to 325 degrees F (165 degrees C). Grease an 8-inch baking pan.

Step 2
Whisk cocoa powder, flour, espresso powder, and salt together in a bowl.

Step 3
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.

Step 4
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.

Step 5
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.