Add warm milk, 1 teaspoon sugar and yeast to a measuring cup or bowl. Stir. Let sit for 10 minutes, until foamy.
Add to a large mixing bowl with shortening, vanilla, eggs and remaining ½ cup sugar. Mix using an electric mixer fitted with a dough hook on low, just until combined. Add 2 cups flour and salt. Mix for a few minutes on low.
With mixer running, add remaining flour 1 cup at a time until 5 cups have been added. With mixer on low speed, continue to add flour 2 tablespoon at a time until the dough no longer sticks to the bowl. Once it pulls away from the sides of the bowl stop adding flour. Increase speed to medium and knead for about 4-5 minutes, or until smooth. The dough may be slightly sticky.
Form into a ball, placing into a lightly oiled bowl, and cover. Set in a warm place to rise until double, about 1 - 1 ½ hours.
Gently punch down dough. Turn dough out onto a floured non-stick surface. Stretch and roll to ½ inch even thickness. Cut with a floured 3 inch doughnut cutter. Place cut doughnuts on a large lightly floured sheet pan lined with wax paper. Cover loosely with a cloth. Let doughnuts rise again until doubled, about 45 minutes.
To make glaze. Melt butter in a large microwave safe bowl. Stir in confectioners sugar and vanilla mixing until smooth. Stir in hot water 1 tablespoon at a time until the icing is thin, but not watery. Set aside.
To fry: Heat oil in a deep-fryer or large deep heavy bottomed pot to 340-350°F. Maintain around 235-240°F while cooking. Cook in batches, carefully lowering doughnuts into hot oil. Turn, frying until golden on each side about 1- 1 ½ minutes total.
Remove from hot oil using a slotted spoon or stainless steel spider. Have nearby a pan fitted with a wire rack. Place fried doughnuts on rack.
Immediately dip doughnuts into glaze while hot, and set back onto wire rack to drain off excess. You can repeat glaze, if desired. Let cool and dry. (You can serve warm, if desired.)
Store in an airtight container at room temperature.