In a large bowl, whisk together the flour, cocoa powder, baking powder and salt.
In a separate small bowl, whisk together the eggs, sugar, buttermilk and melted butter. Stir the wet ingredients into the dry ingredients until combined. (The dough will be very wet.) Cover the bowl with plastic wrap and refrigerate it for 2 to 3 hours until chilled.
When you are ready to make the doughnuts, attach a deep-fry thermometer to the side of a large heavy-bottomed pot. Add 3 to 4 inches of oil, ensuring there are at least 2 inches from the top of the oil to the top of the pot. Begin heating the oil over medium-high heat. (The oil is ready for frying when it reaches 375ºF.)
Scrape the dough onto a well-floured work surface. Flour your hands then pat the dough out until it is about 1/2-inch thick all around. (The dough will be very sticky and wet. Do not hesitate to add more flour as needed to prevent the dough from sticking.)
Flour the cookie cutter, lightly flour a small baking sheet and line a separate baking sheet with paper towels. Using the cookie cutter, cut out circles from the dough and transfer them to the floured baking sheet, re-flouring the cookie cutter after each cut. Gather the scraps, pat them down and cut out as many additional circles as you can. Once the oil reaches 375ºF, carefully add the doughnut holes in batches of 3 or 4, cooking them for 1 to 2 minutes per side and flipping them as needed. (See Kelly's Notes.)
Using a slotted spoon, transfer the doughnut holes to the paper towel-lined baking sheet. Allow the doughnut holes to cool completely then dip them into the prepared glaze, shake off any excess and decorate them with sprinkles (optional). Place the doughnut holes on a cooling rack to allow the glaze to set slightly. Serve the doughnut holes immediately or store them in an air-tight container.