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.