Quarter the potatoes and place in a sauce pan. Cover with cold water and bring to a boil over medium heat.
Boil the potatoes for 10-15 minutes or until tender. Drain the potatoes, reserving the potato water.
Push the potatoes through a food mill or mash by hand, leaving no lumps. Measure out 3/4 cup of mashed potatoes and place in the bowl of a stand mixer. Discard remaining potatoes or reserve for another use.
Add 1/2 cup of flour and the egg and mix, using the dough hook, until just combined.
In a small pot, heat 3/4 cup of potato water, sugar, butter, and salt over low heat until the butter has melted.
While the butter is melting, add 2 tablespoons of the warm potato water (between 100-120 degrees) to a small bowl with the yeast and let sit for 5 minutes to bloom.
Add the potato water and butter mixture to the mixer along with the yeast mixture, mace, and nutmeg.
Turn the mixer to low and start adding the flour, 1/2 cup at a time, until a firm dough forms. The dough should clear the sides of the bowl. Continue kneading until smooth and elastic, 5-10 minutes.
Remove dough to a greased bowl and place in a warm spot to rise until doubled in bulk, about one hour. We like to proof our dough in the oven with the light turned on to keep it a bit warmer than room temperature.
Once dough has doubled in size, punch the dough down and turn onto a floured surface. Roll to 1/2 inch thickness.
Cut the dough with a floured biscuit or donut cutter.
Cut small pieces, 3-4 inches each, of parchment paper and arrange on baking sheets. Dust lightly with flour. Place cut donuts on the parchment sheets to help make transferring the dough to the oil easier. This will help you from deflating the dough during transfer.
Transfer the cookie sheets to the oven or cover with a tea towel to rise. Let rise until doubled, about 45 minutes to 1 hour.
Heat a deep skillet or pot with 2 inches of oil to 375 degrees.
Gently slide the donuts off the parchment sheets and into the hot oil. Fry the donuts in batches (don't overcrowd the pan), flipping as needed until golden brown on both sides, about 2-3 minutes.
Let donuts cool on a paper towel lined plate until easy to handle.