In a small pot, boil the potatoes in at least 2 cups of water. Remove and reserve 3/4 cup of the potato cooking water. Drain and mash the potatoes until smooth.

In a medium bowl, stir together the mashed potatoes, reserved potato cooking water and sugar. Allow to cool a few minutes until it is closer to room temperature and whisk in the yeast. Cover with a tea towel and let the mixture rest for 15-20 minutes. It should look foamy and like the yeast is starting to do something. If it doesn’t, start over.

Meanwhile, melt the butter in a small saucepan and set aside to allow to cool slightly.

Whisk the potato mixture into a large bowl or bowl of a stand mixer through a metal sieve.

Stir in the egg. Stir 1 cup of the flour until the egg looks mixed in, about a minute. Add the second cup and mix until dough starts to form around the dough hook (or in your bowl around your spoon if mixing by hand), about 3-5 minutes. Stream in the melted butter as the mixer goes (or mix it in by hand a little at a time). Mix 1 minute then slowly add the remaining 1 cup of flour in as the mixer is going with the dough hook (or use a spoon and some muscle) until the dough forms a large ball and has completely come away from the sides of the bowl. Remove to a larger, buttered bowl that has room for the dough to double in size. Cover with a tea towel and put in a warm place to rise for 3 hours.

On a clean, floured surface, roll out the dough into an 1/2-inch-thick rectangle. Cut into large 3x3” squares. Place each on a large baking sheet or platter at least an inch apart. Make a little dimple with your finger tip in the middle of each doughnut but don’t push all the way through. Cover with a tea towel and allow to rise again in a warm place for 30-45 minutes.

Heat a few inches of oil in a large walled skillet or Dutch oven to 365°. Meanwhile, line some platters or cookie sheets with paper towels. Pour some more granulated sugar in a shallow bowl or plate.

Working in batches, fry the doughnuts until golden brown on both sides, flipping once, about 2 minutes per side. Drain briefly on paper towel lined plates. (see note for details about how I did this myself)

Dip both sides of each doughnut in granulated sugar and serve.

Best served the same day they are made.