In a large bowl, dissolve yeast and 1 teaspoon sugar in warm water. In another bowl, combine milk and poppy seeds. Let each bowl stand for 5 minutes. Add the potatoes, eggs, shortening, remaining ½ cup sugar, salt, cinnamon, poppy seed mixture and 3 cups flour to the yeast mixture. Beat until smooth. Stir in enough remaining flour to form a soft dough.
Turn onto a floured surface; knead until smooth and elastic, about 6-8 minutes. Place in a greased bowl, turning once to grease top. Cover and let rise in a warm place until doubled, about 1 hour.
Punch dough down. Turn onto a lightly floured surface; divide into fourths. Roll out each portion to 1/2-in. thickness. Cut with a lightly floured 2-1/2-in. doughnut cutter. Place on greased baking sheets. Cover and let rise for 30 minutes.
In an electric skillet or deep-fat fryer, heat oil to 375°. Fry doughnuts, a few at a time, for 1-1/2 minutes on each side or until golden brown. Drain on paper towels.
In a saucepan, bring sugar, milk and butter to a boil. Cook and stir for 1 minute. Remove from heat; cool completely. Stir in confectioners' sugar, salt and vanilla until smooth. Drizzle over doughnuts.
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