Dissolve yeast in warm water. Combine milk, shortening, sugar and salt. Cool until lukewarm. Stir in dissolved yeast, potatoe and eggs. Gradually add enough flour to make a soft dough. Turn onto floured surface; knead until smooth and satiny. Place in lightly greased bowl; turn over to greased top. Cover.
Let rise in warm place until doubled, 1 to 1-1/2 hours. Roll to 1/2 inch thickness; cut with 3-inch doughnut cutter. Cover; let rise until doubled. Meanwhile, stir confectioners sugar, water and vanilla together. (mixture will look like very thich cream.)
Fry doughnuts in deep hot fat (375 degrees F.) until lightly browned. Drain on absorbent paper. Drop doughnuts into sugar glaze. Turn to glaze on all sides quickly and place on cooking rack until glaze is set