Boil potato quarters:

Cut the potatoes into quarters and put them in a saucepan. Add cold water to cover and bring to a boil over medium heat. Cook for 10-15 minutes until soft. Drain, saving the potato water.

Mash and measure potato:

Put the potatoes through a food mill or mash manually until completely smooth. Spoon out 3/4 cup of mashed potato into the bowl of a stand mixer. Use or discard the remainder.

Mix potato, flour, egg:

To the mashed potato, add 1/2 cup (60 g) of the flour and the egg. Blend with the dough hook until barely combined.

Warm potato water mixture:

In a small saucepan, warm 3/4 cup of the reserved potato water together with the sugar, butter, and salt over low heat until the butter liquefies.

Bloom yeast in water:

While the butter melts, combine 2 tablespoons of warm potato water (100-120°F (50°C) / 38-49°C) with the yeast in a small bowl. Allow to sit for 5 minutes until foamy.

Combine mixtures with yeast:

Pour the potato water-butter mixture into the mixer bowl, then add the bloomed yeast, mace, and nutmeg.

Knead dough until elastic:

Set the mixer to low speed and incorporate the remaining flour in 1/2 cup (60 g) increments until a stiff dough forms that pulls away from the bowl sides. Knead for 5-10 minutes until smooth and elastic.

First rise until doubled:

Move the dough to a greased bowl, cover, and let it rise in a warm place (e.g., oven with light on) until doubled in size, about 1 hour.

Roll dough to half-inch:

Deflate the dough and turn it out onto a floured surface. Roll to 1/2-inch thickness.

Cut donut shapes:

Use a floured biscuit or donut cutter to cut out shapes. Arrange the cut donuts on small squares of parchment paper (3-4 inches each) placed on lightly floured baking sheets. This avoids deflating during transfer.

Second rise until doubled:

Cover the baking sheets with a tea towel or put them in a turned-off oven and let rise until doubled, about 45 minutes to 1 hour.

Heat oil to 375°F:

Pour 2 inches of vegetable oil into a deep skillet or pot and heat to 375°F (190°C).

Fry donuts until golden:

Carefully slide the donuts off the parchment into the hot oil. Fry in batches without crowding, turning as needed, until golden on both sides, about 2-3 minutes.

Cool donuts on towels:

Let the donuts cool on a plate lined with paper towels until they are cool enough to handle.

Make vanilla glaze:

For the glaze: warm the milk in a small saucepan until hot. Take off the heat and mix in the powdered sugar and vanilla. Whisk until smooth.

Glaze and set donuts:

Set a wire rack over a sheet pan. Dip the warm donuts into the glaze and place on the rack. Allow to set for 5-10 minutes until the glaze hardens. Serve right away.