In a bowl of your stand mixer, wit a paddle attachment, stir together the dry ingredients. Add to bowl, egg, milk, melted butter, and vanilla. Mix to combine. Cover and let rest for 5 minutes.
Switch to a dough hook attachment, and knead the dough for 6 to 8 minutes, until it's smooth and soft. You can also knead the dough by hand, but it will take longer to knead. Place the dough in a lightly greased bowl, turn it over to coat the top,. Cover with a clean kitchen towel and let it rise for 1 1/2 to 2 hours, until doubled in size.
Gently deflate the dough by pressing your fist down into the dough. Turn the dough onto a lightly floured surface. Using a lightly floured rolling pin, 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 about 30 minutes to an hour, until doubled in size. Heat the oil or shortening in a heavy frying pan or skillet to 350°F. Set up a wire cooling rack, and place paper towels over it (this will be used to cool and drain the cooked donuts when you take them out of the oil). Carefully place the doughnuts in the oil and fry until golden brown, about 1 minute, then flip over and cook on the second side for no longer than a minute. Cook only 2 to 3 donuts at a time, otherwise you will crowd the pan and the oil temperature will drop. Remove from the oil with a slotted spoon and place on the cooling rack.
In a small mixing bowl combine the glaze ingredients. Dip each doughnut into the glaze. Coat on each side.