Scrub your potato, quarter, and add enough water to cover potatoes by 2". Do not add salt. Bring to a boil and cook until softened, about 10 minutes. Remove the potato and mash. Set aside 1 cup of mashed potatoes. Pour potato water into a measuring cup, and set aside 1 ½ cups of potato water to cool.
In a small bowl or measuring cup, sprinkle yeast over 1/4 cup warm water. Add a pinch of sugar to test the active yeast.
In a large mixing bowl, combine potato water, sugar, salt, and yeast mixture. If using a stand mixer, add to the mixing bowl and use the dough hook attachment. Continue mixing while adding the ingredients: butter, and mashed potato.
Add flour 1 cup at a time, mixing thoroughly between cups. Stop adding flour after the 6th cup and see if the dough cleans the sides of the bowl. If not, add 1/2 cup of flour at a time until the dough forms a ball. Do not add more than 7 cups of flour. Knead dough in a stand mixer for at least 5 minutes or by hand for 10 minutes.
First Rise
Place dough in a lightly greased bowl to rise. Lightly grease the top of the dough and cover it with plastic wrap or a towel. Place dough in a warm place and let rise for 2 hours (or until doubled in size).
Second Rise
After first rising, punch the dough down. Begin the second rise for two hours or until double in size, or refrigerate the dough overnight. If refrigerated overnight, place dough in a warm spot the next morning and let rise for 2 hours.
Then, 2 hours before baking, divide the dough into two pieces and continue dividing the dough into 36 similar-sized balls. Shape balls by rolling them by hand.
Last Rise
Place rolls on greased cookie sheet. Cover and let rise in a warm place until double in size.
Preheat oven to 400ºF.
Bake in preheated oven for 12-15 minutes. Butter tops of each roll after baking to preserve soft crust. Serve warm.
Notes
Although this recipe calls for all-purpose flour, I have used both Kamut flour and bread flour with great results.