I have made this recipe (which was passed down from my mother to me in 1950) for my children and my grandchildren. It is really what got me interested in cooking to begin with. And as they say, "the rest is history".
Activate yeast in the work bowl of a large stand mixer. Add dry yeast to 1/4 cup warm water (about 105° F.) Set aside for 10 minutes.
Meanwhile, scald milk. Cool to lukewarm.
With flat beater in place add lukewarm milk, sugar, salt, shortening and eggs to yeast in mixer bowl. Mix on low until ingredients are blended. Switch to dough hook and add flour 1 cup at a time, stirring until mixture forms a ball of dough which follows the dough hook around the bowl.
Place dough into a lightly greased bowl and cover with a damp tea towel or oiled plastic wrap. Allow to rise in a warm place for about 30 to 60 minutes.
Punch dough down to exhaust air. Place onto a barely oiled surface. Roll 1/2 inch thick and cut with a doughnut cutter. Re-cover doughnuts with clean tea towels or plastic and allow to rise until doubled in bulk. (45 minutes to 1 hour).
Using caution, heat oil to 350° F. , in a deep skillet or Dutch oven (I use a Fry Daddy) until a speck of dough dropped into hot fat dances on the surface. Fry doughnuts, three or four at a time, until golden brown, (about 1-1/4 to 1-1/2 minutes per side), turning as necessary. Drain on inverted cooling rack placed over paper towel - lined sheet pan.
TO PREPARE GLAZE: Mix boiling or very hot tap water with powdered sugar. Beat well until smooth. Beat in vanilla. Glaze doughnuts while still warm. (NOT HOT) !!
NOTE: You are aiming for a very slightly slack dough because one that is too dry, will make a heavy end product.