Preheat oven to 350º F. Grease a 12-cavity donut pan and set aside.
Sift together flour, sugar, cocoa, baking soda, baking powder and salt in a large mixing bowl.
Add milk, pumpkin, egg, vanilla, and butter to the flour mixture and mix well with a whisk.
Evenly distribute the batter into each of the 12 donut cavities. Use a small spatula or your index finger to gently swirl the batter in each cavity to ensure even distribution.
Bake donuts for 10 minutes or until a toothpick inserted in the center comes out clean.
Allow donuts to cool in donut pan for five minutes, then add donuts to a cooling rack to fully cool.
While donuts cool, make the glaze. Start by sifting together the powdered sugar and cocoa powder, then adding the milk. Combine until well mixed.
Place the top of each fully cooled donut, one-by-one, into the glaze to coat the top of the donut, then place glazed donut back on the cooling rack and allow the glaze to drip off of it. (Put a piece of parchment paper under the cooling rack to catch glaze drip off.)
Allow glaze to set (about 1 hour) before enjoying. Store donuts in a sealed container for up to three days. Donuts freeze well too.
Make sure the donuts are fully cooled before glazing, otherwise the glaze will melt upon contact.You can use pumpkin or applesauce in this recipe, although some have told me yogurt works well too. If you want to add sprinkles to your donuts, add them while the glaze is wet (before it sets). Lightly sprinkle the sprinkles on top of the glaze.Don't like glaze? Perhaps add a dusting of powdered sugar instead?