Preheat oven to 325°. Spray a 6-cavity donut pan with cooking spray.
To make the donuts, whisk the flour, sugar, cocoa, baking powder, baking soda, and salt in a large bowl. Stir in milk, coffee, egg, vanilla, and oil. Stir until just combined. Place batter in a large ziploc bag with the tip cut off. Pipe into donut pan, filling each cavity 3/4 full. (You can probably get 7 donuts, but I was lazy and didn’t want to reuse my pan, so I only made 6.)
Bake for 9-10 minutes. Allow to cool for five minutes, and then remove from pan.
While donuts are cooling, make the glaze. Mix the milk, coffee, vanilla, and sugar in a small saucepan. Heat on low and stir until the ingredients are all combined. Turn off heat. Carefully dip the tops of each donut in the glaze. Place on a wire rack that is on top of a cookie sheet to catch the drips. Let sit for about a minute. Stir the glaze in the pan, reheating if necessary. Re-dip the donuts, on the bottom this time, and dipping around the edges so the donut is entirely covered. You may want to use a fork in the center of the donut for this step so you don’t burn your fingers! Let sit on the wire rack until cooled.
Make the coffee frosting by mixing the butter, powdered sugar, and brewed coffee with a hand-held mixer on low speed until thick and creamy. Once glaze is somewhat dry (it will remain a little sticky) frost the donuts and sprinkle with chocolate sprinkles. Serve with a cup of coffee for a real jolt!