In a large bowl, mix the milk, sugar, shortening, baking soda, baking powder, vanilla, yeast, warm potato water (water that potatoes have been boiled in), and mashed potatoes.
Mix until well combined. I used a wooden spoon to mix it all together.
Let the dough set for about 30 minutes or until it's a little foamy with bubbles all around the edges.
Once foamy, add the beaten eggs, salt, and flour.
Knead with your hands until well combined.
Once kneaded, let rise until double in size (about an hour).
Roll out about 3/4" thick, cut as for doughnuts (we used two different sizes of circle cookie cutters).
Let rise until double in size.
Cook in hot vegetable oil, flipping when one side is golden.
While still hot dip in glaze.
How to make the glaze: In a medium bowl, whisk together powdered sugar and cocoa powder.
Slowly stir in milk and vanilla extract.
Whisk until silky and smooth. If you need a touch more milk to make this a dippable glaze, add a bit more.
Dip doughnuts in chocolate glaze and let rest to harden slightly.