Place warm minced or mashed potato in the bowl of your stand mixer. Stir in softened butter, warm milk, yeast, sugar and salt.
Begin kneading the flour into the wet ingredients on low speed for about 10 minutes until you have a soft round ball of dough. Transfer dough to a lightly oiled bowl and cover with plastic wrap or a clean towel. Let rise for about 1 1/2 hours or until doubled in size.
Line a baking sheet with parchment paper. On a lightly floured clean surface, divide dough into 8 equal sized smooth round balls. Transfer to the baking sheet, cover with a clean towel and let rise another hour or so until dough has doubled in size.
Preheat oven to 375 degrees. In a small bowl, whisk together an egg and a teaspoon of water. Brush the tops of the buns with egg wash. Transfer buns to the oven and bake for about 15 minutes until tops are golden. Remove buns from the oven and transfer to a cooling rack. Enjoy!