Place cubed potatoes in a saucepan, cover with water, and boil. Cook until tender, about 10 minutes. Reserve 1 cup of cooking water, then drain and mash until smooth.
Stir in the mashed potatoes, melted butter, egg, and salt until well combined.
Gradually incorporate the flour, mixing until a soft dough forms. It is okay if the dough is slightly tacky. If needed, add a little more flour to the dough and flour your hands while working with it.
Cover the buns with a towel and let rise until puffy, about 30-50 minutes. Meanwhile, preheat the oven to 350°F.