Soft and Fluffy Glazed Donuts

Sprinkle the yeast over the warm water, and let stand for 5 minutes, or until foamy.
In a stand mixer fitted with a paddle attachment, combine the yeast mixture, milk, sugar, salt, eggs, shortening, and 2 cups of the flour. Mix for a few minutes at low speed. Add the remaining flour 1/2 cup at a time, until the dough no longer sticks to the bowl.
Switch to a dough hook and knead the dough for about 5 minutes, or until smooth and elastic. Place the dough into a greased bowl, and cover it. Set in a warm place to rise until it has doubled in size. The dough is ready when you can press it and it creates an indentation.
Turn the dough out onto a floured surface, and roll it out to 1/2 inch thickness. Cut out the donuts, cover them with a cloth and leave them to rise again until they've doubled in size.
To make the glaze, melt butter in a saucepan over medium heat. Add the confectioners' sugar and vanilla and stir until smooth. Remove the mixture from the heat, and stir in hot water one tablespoon at a time until the icing is somewhat thin, but not watery. Set the glaze aside while you fry the donuts.
Heat oil in a deep-fryer or large heavy skillet to 350°. Place the donuts in the hot oil using a wide spatula. Fry doughnuts on each side until golden brown. Drain the fried donuts on a wire rack.
To glaze, dip donuts into the glaze while they are still hot, and set onto wire racks (with a baking sheet or parchment underneath to catch the drips) to drain off excess.