Combine yeast, 1 teaspoon sugar, and lukewarm water in a small mixing bowl; stir well, and let stand 5 minutes or until bubbly.
Combine scalded milk, potatoes, shortening, remaining sugar, and salt in a large mixing bowl; stir until shortening melts. Cool to lukewarm (105° to 115°); stir in yeast mixture and 2 cups flour. Beat until smooth. Cover and let rise in a warm place (85°), free from drafts, 1 hour or until doubled in bulk.
Stir dough down. Add eggs and enough remaining flour to make a soft dough. Turn dough out onto a floured surface; knead 8 minutes or until smooth and elastic. Place dough in a large greased bowl, turning to grease top. Cover and refrigerate overnight.
Punch dough down. Shape into 1 1/2-inch balls; arrange in 2 lightly greased 13- x 9- x 2-inch baking p>ans. Brush rolls with melted butter. Cover and repeat rising procedure 1 hour or until doubled in bulk. Bake at 400° for 12 to 15 minutes. Remove from pans, and serve warm.