If you don't have leftover plain mashed potatoes, you will need to make them. I like to use fresh russet potatoes, but you can use instant potato flakes. I recommend pressing them through a potato ricer to remove all lumps.
In a large bowl, combine the yeast, warm water, and about a tablespoon of the sugar. Let sit for about 10 minutes until bubbly.
Add the remaining sugar, oil, mashed potatoes, milk, eggs, salt, nutmeg, and about two cups of the flour. Beat until smooth with an electric hand mixer.
Stir in enough flour to make a soft dough that is slightly sticky but doesn't stick to the sides of the bowl.
Dump the dough out on a lightly floured surface and knead for a couple minutes or until the dough is smooth.
Spray the bowl with cooking spray and place the dough in the bottom. Let sit in a warm place until the dough doubles in size, about an hour.
Spray the counter with cooking spray and use a rolling pin to roll the dough to about 1/2" thick. Cut with a floured doughnut cutter. Place on cookie sheets lined with silicone liners or parchment paper. Continue
Let the donuts rise until doubled, 30-45 minutes. Heat oil to 360° in a pot or skillet. It needs to be at least 2 inches deep. (I use the T-Fal fryer and love it!) Cook donuts for about 45-60 seconds on each side, or until the outside is golden brown and the inside is cooked. Use a chopstick to flip and remove the donuts from the hot oil. Don't overcrowd the pan, just cook a few donuts at a time to maintain oil temperature. (If your donuts are thicker, you may need to increase the fry time.)
Place the cooked donuts on paper towels and allow to cool slightly before glazing.
For glaze: Combine all ingredients in a shallow bowl. Whisk together until smooth, adding enough milk to make a thin glaze.
Place a wire rack inside a baking sheet. Let the donuts cool sightly before dipping them in the glaze. I dip both sides of the donuts, but you can just dip the top if you prefer. After dipping, place them on the cooling rack to allow excess glaze to drip off.