Preheat oven to 350 degrees F (180C). Lightly coat donut pans with oil/butter.
In a large bowl, mix together all-purpose flour, baking soda, salt, cocoa powder, and sugar. Sift them in order to mix them easily with wet ingredients. Set aside.
In a medium bowl, whisk together the egg, buttermilk, oil, vanilla, red food coloring, and vinegar.
Slowly pour the wet ingredients into the dry ingredients, whisking them together until all the clumps are gone. Batter will have a very heavy and wet texture.
Using an icing bag (or a zip lock bag with the corner cut out), pipe the donut mix into the donut pan. You can also spoon the donut mix in, but it will be messier and harder to get an even fill. Fill the donut rings to about 3/4 full. Bake for about 15-17 minutes.
Let donuts cool in the pan for 10 minutes, then gently remove them and place on a wire rack to cool completely. Drizzle with the icing.
For the Vanilla Cream Cheese Icing
Combine cream cheese, powdered sugar, and vanilla in a small bowl. Use a spatula to press the ingredients around until combined. Add the milk a little at a time to get the desired consistency.