Preheat the oven to 350°F. Lightly greasy two doughnut pans.
In a large bowl, whisk together the cocoa powder, flour, sugar, baking powder, baking soda, and salt. In another bowl whisk together the eggs, milk, vanilla, and vinegar. Add to the dry ingredients with the melted butter and stir until combined.
Scoop the batter into a piping bag, cut a hole in one corner, and pipe the batter into the doughnut molds, filling about 3/4 full. Bake for 12 to 15 minutes, or until a toothpick inserted into the center comes out clean. Let cool in pans for 2 minutes before flipping the pans upside down over a rack to remove the doughnuts.
For the Icing
Whisk together the sugar, milk, and vanilla until smooth and combined. The mixture will be very thick. Dip the tops of the doughnuts in the icing or use a spoon to spread a thick layer.
Sprinkle with crushed Oreo cookies before letting set.