Make the Dough
Steam or bake the potato with the skin on until fully tender, about 30 minutes. Peel the potato while hot and push it through a sieve to finely mash it. Measure out ¾ cup (6 oz/170g) and place in a large bowl. This is the amount you will need for the recipe.
To the bowl of potatoes, add the milk, butter, egg, and vanilla and whisk until smooth.
In the bowl of a stand mixer fitted with a dough hook, whisk together the flour, sugar, yeast, and salt, then pour in the wet ingredients.
Mix on medium-low speed for 6 to 8 minutes. The dough should be very soft but still be able to mostly clear the sides of the bowl and just hold its shape.
Transfer the dough to a large, oiled bowl, cover and let rise in a warm place until doubled, about 1 ½ hours.
Line a baking sheet with parchment paper and sprinkle some flour on top.
On a lightly floured surface, pat out the dough to ½ inch (12 mm) thick and using a 3-inch (7 ½ cm) and 1-inch (2 ½ cm) cookie cutter, cut out doughnut shapes. Transfer to the prepared baking sheet and gather and reroll scraps to get 12 doughnuts.
Cover the doughnuts with a second baking sheet and let the doughnuts rise a second time until doubled, about 40 minutes.
Make the Glaze
Whisk the powdered sugar, milk, and vanilla extract together to make a smooth, thin glaze that should go on fairly clearly. Add more milk if needed. Set aside.
Fry the Doughnuts
In a large, heavy-bottomed high-sided pan, pour oil 2 to 3 inches deep and heat on Medium to Medium-Low for 5 minutes. (You can also use a candy thermometer and heat to 325°F (165°C).
While the oil is heating, place a wire rack on a baking sheet next to the heating oil to drain the doughnuts after frying.
Once the oil has reached temperature, fry the doughnuts, 2 to 3 at a time. After about a minute, when the doughnut is golden on one side, flip and fry the other side.
Rest on the wire rack to cool and drain while you fry the remaining doughnuts.
While the doughnuts are still hot, dip in the glaze, letting the excess drip back into the bowl. Return to the wire rack to let them set.
Serve right away while still warm. These doughnuts are best eaten the day they are made.