In a saucepan with boiling water, add the potato to cook until fork tender, about 15-20 minutes.
Add ¾ cup of the potato water to the bowl of a stand mixer fitted with the whisk attachment. Once cooled until warm to touch, add the yeast and sugar to bloom for 5-10 minutes.
Add 1 cup mashed potato, egg, and salt to beat for 1 minute.
While on medium speed, add the softened butter a tablespoon at a time.
Using the hook attachment, add 3 cups of the flour to mix on LOW speed until incorporated. If the dough is too sticky, add the remaining ½ cup of flour.
Knead for 5-6 minutes, until a smooth and pliable dough.
Place in a greased bowl to rise in a warm environment until doubled in size, about 1 hour.
Shape the risen dough into a 9” log. Place in a grease 9x5” loaf pan lined with parchment paper.
Cover to rise 30 minutes to 1 hour, until doubled in size.
Preheat oven to 350F.
Bake the potato bread for 20-25 minutes.
Cover with aluminum foil, then bake an additional 20-25 minutes, until deeply golden.
Let cool at least 30 minutes before slicing.