To make the dough: In a large bowl or the bucket of a bread machine set on the dough cycle, whisk together the dry ingredients. In a small bowl, whisk together the egg, milk, melted butter, and vanilla. Add all at once to the dry ingredients. If you’re using a bread machine, press Start. If you’re preparing by hand or mixer, mix and knead to make a soft dough. Cover and let rest for 5 minutes.
If preparing by hand or mixer, knead the dough after its rest for 6 to 8 minutes, until it’s smooth and soft. Place the dough in a greased bowl, turn it over to coat the top, cover, and let it rise for 1 1/2 to 2 hours, until doubled in bulk.
To shape the doughnuts: Deflate the dough and turn it out onto a lightly floured surface. Gently roll it 1/4″ thick, and cut out doughnuts with a 3-1/2″ donut cutter. (if you don’t have a donut cutter – then use a round cookie cutter or biscuit cutter, then a smaller cutter to cut out the hole in the middle – a round piping tip or smaller cutter works great! Cover loosely with greased plastic wrap and let the doughnuts rise for 30 minutes to an hour, until doubled.
To fry: Heat the oil or shortening in a heavy frying pan or skillet to 350°F. Carefully place the doughnuts in the oil, 2 or 3 at a time, and fry until golden brown. Turn over and cook the second side; each side should take no more than a minute. Remove from the oil with a slotted spoon and drain on absorbent paper. Fill or frost doughnuts as desired, using your choice of sugar topping or glaze.
Make The Glaze. Sift together the confectioners sugar and cocoa powder. Stir in milk, vanilla, butter, and corn syrup (if using). Adjust consistency if needed.
Dip Donuts. Dip the top of each donut into the glaze, letting the excess drip off. Place on to a wire rack.
Storage: These are best eaten the day they are made.
Make ahead: After cutting out the donuts, place on a baking sheet, cover and keep in the fridge and fry the next day.
Baking: If you wanted to bake the donuts instead of frying, after the donuts have been shaped have gone through the final rise then bake the donuts for about 14 minutes in a 350oF oven. They won’t be quite as crispy on the outside and get that deep-fried flavor but this process will be much easier.
Yeast- You can use instant or active dry yeast and add it directly to the bowl, no proofing required.
Milk – I use full fat milk for richness, but I think a lower fat like 2% milk would work fine. I make sure my milk has been slightly warmed, and not cold from the fridge.
Corn Syrup – I prefer to not use it, but using it will make your glaze stay nice and shiny.
Cocoa powder – I used Dutch process cocoa powder for a richer, darker chocolate glaze but you can also use natural unsweetened for more of a milk chocolate glaze.