Boil the potato until tender. Drain, reserving ¾ cup of the cooking water. Mash the potato and set aside.
In a large bowl, combine the hot potato cooking water, sugar, butter, and salt. Let stand until the butter is melted and the mixture is warm (105 degrees to 115 degrees F). Stir in the yeast and let stand for 5 to 10 minutes. Beat in the egg and potato by hand. Using a wooden spoon, stir in as much of the flour as you can. Turn the dough out onto a floured surface and knead for about 3 to 5 minutes, adding enough flour to make a moderately soft dough. Place in an oiled bowl, turning to coat all sides with oil. Cover and let rise in a warm place until doubled in bulk, about 1 to 1-½ hours.
Punch the dough down. On a lightly floured surface, roll the dough out to a ½-inch thickness and cut with a 3-inch round cutter. Place the rounds ¼ inch apart on a lightly greased baking sheet. Cover and let rise until doubled in bulk, about 30 to 45 minutes.
Preheat the oven to 350 degrees F. Bake for 18 to 20 minutes, or until golden brown. Remove to a wire rack to cool.