Preheat the oven to 350°F.  , To make the batter: In a large mixing bowl, whisk together the cocoa, flour, sugar, baking powder, baking soda, salt, espresso powder, and chocolate chips. Set aside., In a large measuring cup or medium-sized mixing bowl, whisk together the eggs, milk, vanilla, and vinegar.  , Add the wet ingredients to the dry ingredients, stirring to roughly blend. Once you can no longer see any bits of raw egg, stir in the melted butter. There's no need to beat the batter, just make sure everything is well-combined., Lightly grease the wells of two standard doughnut pans. If you don't have two pans, simply bake the batter in two batches. Spoon the batter into the prepared pan(s), filling the wells about half to two-thirds full; there should be about 64g to 70g batter in each well., Bake the doughnuts for 12 to 15 minutes, or until a toothpick inserted into the center of one comes out clean. Don't over-bake the doughnuts, as this could dry them out., Remove the doughnuts from the oven and after 30 seconds or so, use a nylon heatproof spatula or table knife to carefully loosen their edges. Turn the pan upside down over a rack, gently tap the pan on the rack, then give it a little shake; the doughnuts should fall onto the rack. If one or two stick, use the spatula or knife to loosen them further. Do all of this quickly; the longer you wait, the more chance the doughnuts will stick. , For simple sugar-coated doughnuts, immediately shake the doughnuts in a paper or plastic bag with 1 tablespoon granulated sugar., If you want to ice the doughnuts rather than shake them in sugar, allow them to cool completely before icing., To make the icing: Combine the chocolate chips and milk or half-and-half in a microwave-safe bowl or measuring cup. Heat until the liquid 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, spreading to coat; or simply spread icing on the doughnuts. Garnish with sprinkles or shaved chocolate, if desired., Store doughnuts, loosely covered, at room temperature for a couple of days; freeze for longer storage.