Place potatoes in a saucepan and cover with water. Bring to a boil and cook until fork tender.
Drain, reserving 1/2 cup cooking liquid; cool liquid to 110-115 degrees. 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 it rise again until doubled, about 20 minutes.
Roll out on a floured surface to 1/2" 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.
For glaze, combine powdered sugar, water and vanilla in a bowl. Dip warm doughnuts in glaze. Cool on wire racks.