Preheat the Oven: Preheat your oven to 350°F (175°C). Grease a doughnut pan.
Mix Dry Ingredients: In a large bowl, whisk together flour, cocoa powder, baking powder, baking soda, and salt.
Combine Wet Ingredients: In another bowl, beat together the eggs, sugar, buttermilk, vanilla extract, and melted butter.
Combine the Mixtures: Gradually add the wet mixture into the dry ingredients. Stir until just combined.
Bake the Doughnuts: Spoon the batter into the doughnut pan. Bake for 12-15 minutes or until a toothpick comes out clean.
Cool the Doughnuts: Let the doughnuts cool in the pan for 5 minutes, then transfer to a cooling rack.
Make the Coffee Glaze: In a small saucepan, whisk together powdered sugar, brewed coffee, milk, and vanilla. Stir until smooth.
Glaze the Doughnuts: Dip each doughnut into the coffee glaze. Let the glaze set before serving.