Heat apple juice until warm (100° to 110°F). Add yeast and allow to sit for 5 minutes.
Combine 3 cups flour, sugar and salt in a large mixer bowl. Add butter, egg, potatoes and yeast mixture. Beat for 2 minutes. Add enough remaining flour to form a soft dough.
Knead on a lightly floured surface for 6 to 8 minutes, until dough is smooth and elastic. Shape dough into a ball and place in a greased bowl, turning once to cover. Cover and let rise in a warm, draft-free place until dough is doubled in size, about 1 hour.
Punch dough down. Divide dough into 12 equal portions and form into rolls. Place in a greased 13 x 9-inch pan. Cover and let rise until doubled, about 40 minutes.
Bake for 25 to 30 minutes in a preheated 350°F oven until well browned. Allow to cool on wire rack.
Whole Wheat Variation: Use 1-1/2 cups whole wheat flour in place of 1-1/2 cups all-purpose flour.
Sweet Potato Tip: Cook or bake sweet potato. OR use a can (15 ounces) of sweet potatoes. Drain well and mash enough to measure 1/2 cup.