Sprinkle the yeast over the warm water and let it stand 5 minutes, or until foamy. 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 stiring 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, or until smooth an elastic. Place the dough into a greased bowl and cover. Let rise in a warm place until doubled. Dough is ready if you touch it and the indentation (THE INDENTATION WHAT CAROLINE???)
Turn the dough out onto a floured surface and gently roll out to 1⁄2 inch thick. Cut witha doughnut cutter (I don't have one, so I always improvise with whatever I have that is round!) and let sit out to rise again until double. Cover loosely with a cloth.
Melt butter in a pan over medium heat. Stir in powdered sugar and vanilla until smooth. Remove from heat and stir in hot water one tablespoon at a time until the icing is somewhat thin, but not watery. Set aside.
Heat oil in deep fryer or heavy skillet to 350 degrees. Fry a few at a time, flipping over once golden. Drain on a wire rack, then dip in the glaze or roll in cinnamon sugar.