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Preheat the oven to 350°F. Lightly grease the top and bottom halves of a doughnut hole pan.
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Whisk together the cocoa, flour, sugar, baking powder, espresso powder, baking soda, salt, and chocolate chips. Set aside.
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Whisk together the egg, milk, vanilla, and vinegar. You may notice some curdling of the milk; that's OK.
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Add the wet ingredients, along with the melted butter, to the dry ingredients, stirring to blend.
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Scoop a tablespoon of batter into the cups in the bottom half of the doughnut hole pan (the side without the vent hole) — a level tablespoon cookie scoop works well here.
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Place the top half on the pan, and lock it in place with the clips.
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Bake the doughnut holes for 8 to 9 minutes, or until a cake tester inserted into the center of one comes out clean.
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Remove the pan from the oven, and allow it to cool slightly before removing the clips and the top half.
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Loosen the edges of the holes, if necessary, and gently turn them out onto a rack.
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Immediately shake the holes in the coating, and set aside to cool.
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Grease the pan again, and repeat the process with the remaining batter.
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Store doughnuts, tightly wrapped, at room temperature for several days; freeze for longer storage.