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Preheat the oven to 400 degrees and grease a donut pan with baking spray. Set aside.
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Mix flour, cocoa powder, brown sugar, granulated sugar, baking soda, salt, and espresso powder together until well combined. Set aside.
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Whisk together the egg, coffee, and half & half.
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Make a well in the center of the dry ingredients, and add the wet ingredients. Mix until combined (batter will be fairly thick).
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Add in sour cream, vanilla extract, and melted butter. Mix until combined.
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Pour batter into greased donut pan, filling each donut cavity about 3/4 of the way full (this batter can make about 7-8 donuts).
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Bake for 8-9 minutes, or until the donut 'springs' back when touched. Allow to cool a few minutes before removing from the pan.