Scald milk by bringing it to at least 180 °F(82 °C) then cool it to less than 105 °F(41 °C).
Sprinkle the yeast over the warm water, and let it sit for 5 minutes.
In a large bowl, mix together the yeast mixture, milk, sugar, salt, eggs, shortening, and 2 cups of the flour. Mix for a few minutes at low speed, or stirring with a wooden spoon. Beat in remaining flour 1/2 cup at a time, until the dough no longer sticks to the bowl. Knead for about 5 minutes. Place the dough into a greased bowl, and cover. Set in a warm place to rise until double. Dough is ready if upon touching, indention remains.
Turn the dough out onto a floured surface, and gently roll out to 1/2 inch thickness. Cut with a floured doughnut cutter. Let doughnuts sit out to rise again until double. Cover loosely with a cloth.
Melt butter in a saucepan over medium heat. Stir in confectioners' sugar and vanilla until smooth. Remove from heat, and stir in hot water 1 tablespoon at a time until the icing is somewhat thin, but not watery. Set it aside then.
Heat oil in a deep-fryer or large heavy skillet to 350 degrees F (175 degrees C). Slide doughnuts into the hot oil using a spatula. Turn doughnuts over as they rise to the surface. Fry doughnuts on each side until golden brown. Remove from hot oil, to drain on a wire rack. Dip doughnuts into the glaze while still hot, and set onto wire racks to drain off excess. Keep a cookie sheet or tray under racks for easier clean up.
↑Weight conversions from USDA National Nutrient Database. Original recipe text and ingredient order preserved. Whole milk, 3.25% milkfat, used for conversion, density adjustment not applied to milk or water. Large egg size presumed.
↑This excessive amount of yeast will result in a strong yeast flavor. Because of the high amount of sugar in this dough, osmotolerant yeast is recommended, along with a reduction in the amount used.
↑The absorption of the oil into the fried product is unknown.
↑Powdered sugar not specified as sifted or unsifted, unsifted presumed.