Preheat the oven to 350°F. Lightly grease the wells of two standard doughnut pans.
In a large mixing bowl, whisk together the cocoa, flour, sugar, baking powder, espresso powder, baking soda, salt, and chocolate chips.
In a large measuring cup or medium-sized mixing bowl, whisk together the eggs, milk, vanilla and vinegar. Add the wet ingredients, along with the melted butter to the dry ingredients, stirring until well combined.
Remove the batter to a zip-top plastic bag or piping bag, cut a hole in one corner, and pipe the batter into the doughnut molds, filling them between 3/4 and full. Bake the doughnuts for 12 to 15 minutes, or until a toothpick inserted into the center of one comes out clean.
Remove the doughnuts from the oven and let cool for 2 minutes. Loosen their edges, turn the pan upside down over a rack, and gently let the doughnuts fall onto the rack.
To make the icing:
Combine the chocolate chips and milk in a microwave-safe bowl or measuring cup. Heat until the milk is steaming and starting to bubble. Remove from the microwave, and stir until the chips have melted and the icing is smooth.
Dip the top of each doughnut in the icing; or spread icing on the doughnuts.