Boil the potatoes until fork tender. Set aside the cooking liquid. Push the potatoes through a food mill or mash by hand, leaving no lumps. Measure out 3/4 cup of mashed potatoes.
In the bowl of a stand mixer, combine the mashed potatoes with 1/2 cup of flour and the egg.
In a small pot, heat the 3/4 cup of potato water, sugar, butter, and salt over low heat until the butter has melted. Stir this mixture into the potato mixture and blend until smooth.
Dissolve the yeast in the remaining 2 tablespoons of warm potato water. Add to the potato mixture. Mix in the mace and nutmeg and mix thoroughly.
Add remaining flour a little at a time, with your mixer on low and fitted with the dough hook, until a stiff dough forms. Continue kneading until smooth and elastic.
Remove dough to a greased bowl and place in a warm spot to rise until doubled in bulk, about one hour.
Turn the dough onto a floured surface and roll to 1/2 inch thickness. Cut the dough with a floured biscuit or donut cutter. (I find this dough to be pretty soft, so I use a biscuit cutter since it's easier to move the full circle around than it is to move around the donut shape.).
Cover and let rise until doubled in bulk, about 45 minutes.
Heat a deep skillet or pot with 2 inches of oil to 375 degrees. Fry the donuts in batches (don't overcrowd the pan), flipping as needed until golden brown on both sides.
Drain a paper towel lined plate.
To make the glaze:
Heat the milk in a small saucepan until warm. Remove from the heat and stir in the powdered sugar and vanilla. Whisk together to form a glaze. Dunk the warm donuts in the glaze. Let sit for 5 minutes to set.