Whisk together the dry ingredients. In a separate bowl, combine the egg, milk, butter, and vanilla and stir into the flour mixture, mixing until well combined. Let the dough rest for 5 minutes, then knead for 6 to 8 minutes by hand or mixer until you have a smooth, soft dough. Place the dough in a buttered bowl, turn it over to grease the top, and let it rise, covered, in a warm place for 1 1/2 to 2 hours, until doubled in bulk. Deflate the dough by punching down and turn it out onto a lightly floured surface. Gently roll the dough out to 1/4-inch thickness and cut with a round cutter (or, using a pizza wheel, cut into strips to make crullers. Cover loosely with greased plastic wrap and let rise again for about 1 hour, until doubled again. Place oil or shortening in a heavy pan or deep skillet and heat to 350-degrees F. Place the doughnuts in the oil, two or three at a time, and fry until golden brown. Turn over and cook the second side. This should take no more than a minute for each side. Overcooking will make the doughnuts tough. Drain on paper towels.