Place potatoes in a saucepan and cover with water. Bring to a boil; cook until
tender. Drain, reserving 1/2 cup cooking liquid. Cool to 110 to 115 degrees F.
Discard remaining cooking liquid. Mash potatoes without milk or butter.
In a large mixing bowl, dissolve yeast in reserved cooking liquid. Add mashed
potatoes, milk, oil, sugar, eggs and salt. Add enough flour to form a soft dough.
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; let rise again until doubled, about 20 minutes. Roll out on
a floured surface to 1/2 inch thickness. Cut with a floured 3-inch doughnut cutter.
In an electric skillet or deep-fat fryer, heat oil to 375 degrees F. Fry doughnuts,
a few at a time, until golden brown.
Combine confectioners' sugar, water and vanilla extract; dip warm doughnuts
into glaze. Cool on wire rack.