Combine yeast, 1 teaspoon sugar, and water in a large bowl; stir well. Let stand 5 minutes.
Combine remaining sugar, milk, potatoes, shortening, baking powder, soda, and salt; add to yeast mixture, stirring well. Stir in 2 cups flour. Cover and let rise in a warm place (85°), free from drafts, 1 hour or until doubled in bulk. Add enough remaining flour to make a soft dough.
Shape dough into a ball; place in a greased bowl, turning to grease top. Cover and refrigerate overnight.
Lightly grease muffin pans. Shape dough into 2-inch balls; place 1 ball in each muffin cup. Cover and let rise in a warm place (85°), free from drafts, 1 hour or until doubled in bulk. Using kitchen shears, cut a shallow cross on top of each roll. Bake at 400° for 10 minutes or until golden brown. Brush with melted butter, if desired.