Peel and cube potato. In saucepan, combine potato and the water. Bring to boiling. Cook, covered, for 12 to 15 minutes or until very tender. Drain, reserving 1 cup of hot cooking liquid. Set liquid aside. Mash potato cubes with a potato masher (should have 1 cup). Set aside.
Combine reserved cooking liquid, butter and salt. Cool to 120 degree F to 130 degree F.
In a large bowl, combine 2 cups of the 4-1/4 cups flour and the yeast. Add reserved cooking liquid and eggs. Beat with electric mixer on low speed for 30 seconds, scraping side of bowl. Beat on high speed for 3 minutes. Using a spoon, stir in mashed potatoes and as much of the remaining flour as you can.
On a lightly floured surface, knead in as much of the remaining flour as you can to make a moderately stiff dough that's smooth and elastic (6 to 8 minutes total). Shape into a ball. Place in a greased bowl; turn once to grease surface. Cover; let rise in a warm place until doubled (1 to 1-1/2 hours).
Punch dough down. Turn out onto lightly floured surface. Divide in half. Cover and let rest for 10 minutes.
Shape into loaves or rolls. For loaves, shape each half of dough into a loaf. Lightly dip tops of loaves into additional flour. Place loaves, floured sides up, in 2 greased 8x4x2-inch loaf pans. For rolls, divide each half of dough into 12 pieces. Shape into balls. Lightly dip tops into flour. Place on greased baking sheets. Cover bread or rolls. Let rise until nearly doubled (30 to 40 minutes).
Bake in a 375 degree F oven. For loaves, bake for 35 to 40 minutes or until bread sounds hollow when tapped (if necessary, cover with foil during the last 15 minutes to prevent overbrowning). For rolls, bake for 20 to 25 minutes or until golden. Remove from pans; cool on a wire rack. Makes 2 loaves or 24 rolls.