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 oubled, 1-1 1/2 hours. Roll to 1/2 inch thickness; cut with 3 inch doughbut cutter. Cover; let rise until doubled, 30 minutes. Meanwhile, stir powdered sugar, water, and vanilla together. Fry in deep hot fat (375F). Drain and drop into glaze.
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