Mashed potatoes and their cooking water are both used in the dough. The result is rolls with a smooth-textured grain and a pleasing, slightly tangy flavor.
Cook potato in lightly salted boiling water, covered, about 15 minutes or until tender. Drain, reserving ¾ cup of the potato cooking water. Mash the potato with a fork. You should have 1 cup of mashed potato.
Place the lukewarm potato water in a small bowl. Sprinkle yeast over it and stir until dissolved. In a large bowl, combine butter, sugar, and salt. Add dissolved yeast and mashed potato; stir in 1 cup of the flour. Cover and set aside in a warm place for 1 hour, until the mixture has increased in bulk and is bubbly.
Stir in more flour by cupfuls to make a stiff dough. Knead on a floured board or in an electric mixer with a dough hook until smooth and elastic, about 8 minutes. Transfer to a greased bowl, cover, and set aside in a warm place until doubled in bulk, 45 minutes to 1 hour.
Grease a 9x13-inch baking pan. Punch dough down and divide into 18 equal pieces. Shape each piece into a ball and arrange in the prepared pan. Cover loosely and let rise 30 minutes. Heat oven to 425 degrees.
Bake 12 to 15 minutes, until tops of rolls are a rich golden brown. Remove from oven and brush tops of rolls with melted butter. Serve hot. (If made ahead, rolls can be reheated in a hot oven. They also freeze well.)