In a large bowl combine flour, cocoa powder, baking powder, salt, and baking soda. In another large bowl whisk together eggs and egg yolk. Add sugar, Greek yogurt, buttermilk, and melted butter; whisk until well mixed. Add the flour mixture to the egg mixture. Use your hands to mix together until all of the flour is moistened. Cover and chill dough about 1 hour or until easy to handle.
On a floured surface, roll out dough to a 1/2 inch thickness. Cut dough with a floured 2 1/2-inch donut cutter, rerolling dough as necessary. Fry donuts, two or three doughnuts and donut holes at a time, in deep hot oil (365 degrees F) for 2 to 3 minutes or until golden, turning once. Using a slotted spoon, transfer to paper towels to drain.
In a medium bowl combine powdered sugar and water. Whisk together until smooth. Dip warm donuts and donut holes in Powdered Sugar Icing.