Preheat oven to 350?F. Grease a standard sized donut pan and set aside.

In a large bowl, mix together flour, cocoa powder, brown sugar, baking soda, baking powder, and salt. Set aside.

In a small bowl, whisk egg, milk, vegetable oil, vinegar, and vanilla extract. Add the wet ingredients to the dry, whisking to combine. Gently fold in the crushed Oreos.

Transfer mixture into a pastry bag or large ziplock with a hole cut into the corner. Pipe batter into donut molds, filling 2/3 of the way up the molds. Bake for 10 minutes, or until a knife inserted into the donut comes out clean. Allow to cool 10 minutes before frosting.

While the donuts are cooling, make the frosting. Combine all frosting ingredients in a small bowl, whisking until smooth. The consistency should be thick but still thin enough to form ribbons when you scoop it from the bowl and back down.

Dip each donut into the frosting and top with more crushed Oreos.