Heat oil in a deep-frier to 350 degrees F (175 degrees C).
In a small bowl sprinkle the yeast over the warm water, and let stand for 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.
Beat in remaining flour 1/2 cup at a time, until the dough no longer sticks to the bowl.
Knead until smooth and elastic. Place the dough into a greased bowl, and cover. Set in a warm place to rise until double.
Turn the dough out onto a floured surface, and gently roll out to 1/2 inch thickness.
Cut with a floured doughnut cutter. Gentlely place the doughnuts in the hot oil. Don't crowd the doughnuts.
Melt butter in a saucepan over medium heat. Stir in confectioners' sugar and vanilla until smooth. Remove from heat, and stir in hot water one tablespoon at a time until the icing is at a runny consciencey.
Fry the doughnuts until golden brown on each side. Dip in the glaze while still warm.
Put a cookie sheet under a wire rack. Place the glazed doughnut on the wire rack and let the excess glaze drip off.