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 (or cool bath water temperature).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 double, about 20 minutes.
Roll out on a floured surface to 1/2.
inch in thickness. Cut with a floured 3-inch doughnut cutter.
In an electric skillet, heat oil to 375 degrees. Fry doughnuts, a few at a time, until golden brown on both sides. Drain on paper towels.
Combine confectioners sugar, water and vanilla in a bowl. Dip warm doughnuts in glaze. Cool on wire racks.