Start by peeling your russet potato and cutting it into chunks. Place the potato in a pot of salted water and bring it to a boil. Cook until tender, about 15-20 minutes. Drain and let it cool slightly before mashing it until smooth.
In a small bowl, combine the warm water and a pinch of sugar. Sprinkle the active yeast over the top and let it sit for about 5-10 minutes, or until foamy.
In a large mixing bowl, combine the mashed potato, whole milk, remaining sugar, and coarse salt. Mix until well combined. Add the foamy yeast mixture and the egg, stirring until combined.
Gradually add the all-purpose flour, one cup at a time, mixing until a dough forms. Knead the dough with a dough hook on medium speed for about 5-7 minutes, or by hand for about 8-10 minutes until smooth and elastic.
Once the dough is smooth, add the softened unsalted butter. Knead it into the dough until fully incorporated.
Place the dough in a lightly greased bowl and cover it with a kitchen towel. Let it rise in a warm place until it doubles in size, about 1-2 hours.
Once risen, punch down the dough and turn it out onto a floured surface. Divide the dough into equal portions, about 2-3 ounces each. Shape each portion into a round bun and place them on a baking sheet lined with parchment paper.
Cover the shaped buns with a kitchen towel and allow them to rise again for about 30-45 minutes, or until puffy.
While the buns are rising, preheat your oven to 375°F (190°C).
Once the buns have risen, bake them in the preheated oven for 15-20 minutes, or until golden brown.
Allow the buns to cool on a wire rack for at least 10 minutes before serving.