Dissolve yeast in warm water. Combine milk, shortening, sugar and salt. Cool until lukewarm. Stir in yeast, potatoes and eggs. Gradually add enough flour to make soft dough. Turn onto floured surface; knead until smooth and satiny. Place in lightly greased bowl; turn over to grease top. Cover; let rise in a 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 (30 minutes).
Meanwhile, stir confectioners sugar, water and vanilla together.
Fry doughnuts in deep hot fat (375°). Drain on absorbent paper. Drop hot doughnuts into glaze. Place on cooling rack until glaze is set. Makes about 3 1/2 dozen.
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