Have the mashed potatoes ready to go. I usually use ⅓ cup of potato flakes rehydrated with a generous ⅓ cup of boiling water which makes about 100 grams of dry-ish mashed potatoes. Otherwise, mash potatoes using your favorite method and weigh out 100 grams.
Meanwhile, in a microwave-safe glass measuring cup, scald the milk. Stir in 1 tablespoon of the sugar. Let milk cool to 110 to 115 degrees F. Sprinkle yeast over warm milk and set aside to proof.
In bowl of stand mixer, combine 1 cup of the flour with the remaining 3 tablespoons sugar, salt, egg, butter or shortening and mashed potatoes, nutmeg and vanilla. Stir in the bubbly yeast mixture and a second cup of the flour. Beat by hand or with the paddle attachment to make a very sticky batter. put the bowl on the stand if you haven't already.
Attach dough hook to the mixer and begin kneading the dough on medium speed. Gradually, about 2 tablespoons at a time, add the reserved ½ cup of flour. Continue adding it gradually and kneading until the dough clings to the hook and is slightly elastic. You may use the full ½ cup, you may stop at ⅓. Go by the feel of the dough and stop to scrape often. Continue kneading with the hook until you have a smooth dough that no longer clings to the side of the bowl, but is not dry. My favorite descriptor is "soft as a baby's butt".
Put dough in a well-greased bowl, roll it to slick the sides, then cover with plastic wrap and let double at room temperature for 1 ½ hours (or 45 minutes if using instant yeast). Punch down the dough. Return dough to bowl, cover with plastic wrap and chill overnight or for 8 to 12 hours.