To make the dough: In a large bowl or the bucket of your 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 2 1/2" to 3" round cutter. 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.