In a small saucepan, combine the butter, potato water, and milk over medium low heat. Heat until butter is melted and mixture is steaming. Remove from heat and let sit 5 minutes, or until mixture reaches 110-115°F.
In a large bowl, combine the potato water mixture with the mashed potatoes. Add the sugar, salt and eggs and mix with a wooden spoon until combined. Sprinkle the yeast over the mixture and stir until incorporated. Add the flour 1 cup at a time, stirring with the wooden spoon, until the dough is smooth and only slightly sticky.
With floured fingers, gather the dough into a ball and place in a greased bowl. Cover with greased plastic wrap and let rise in a warm in place until doubled, about 1 hour.
Punch down dough, then pat it out into a rectangle on a floured surface. Flour your fingers as necessary. Cut the dough into 2-3 inch squares. Shape each piece into a ball by pinching all four sides underneath to seal. Place buns almost touching in a greased pan. Cover with plastic wrap and let rise until doubled, about 30 minutes.
Brush rolls with melted butter. Bake at 400°F for 15-20 minutes until golden. Remove from oven and brush with butter. Serve warm or at room temperature. Store in an airtight container at room temperature for several days.
Note: To make mashed potatoes, simmer peeled, cubed potatoes until fork tender. Drain and mash with a vegetable masher or pass through a ricer. Let cool completely.