Make the cinnamon sugar mixture: Place 8 tablespoons sugar and 1 tablespoon cinnamon in a small bowl and toss to combine. Set aside.
Make the donuts: Bring a saucepan of water to boil over high heat and add the potatoes with the peels still on. Cook for 20 minutes or until tender. Let potatoes cool slightly and then mash in a large bowl with skins on until no chunks remain. Measure mashed potatoes. Keep 1 cup in the bowl and discard any extra (i.e. eat any extra).
Add the eggs and milk to the potatoes and stir well to combine.
In a separate, smaller bowl, whisk together 1 cup sugar and baking powder to combine. Stir into potato mixture, followed by the salt, shortening/butter, and vanilla.
Add the flour, about 1 cup at a time, until a soft dough forms (I used all the 4 cups, but you may need to use less). The dough should be just slightly sticky, but easy to roll. Do not knead the dough.
Heat 5 inches/12 cm of oil in a cast-iron pot to medium-high, or about 350 degrees F.
While oil heats, roll out dough on a floured surface about ½-inch thick. Cut out donuts with a donut cutter, the size you want, and carefully add a few at a time to the hot oil. To prevent oil from splashing and burning you, do not drop the donuts on the oil from more than 1 to 2 inches above the oil. Once the bottom of donuts is golden brown, turn to the other side. Drain any of excess oil on a paper-towel lined plate and then immediately dredge hot donuts in cinnamon sugar mixture.