Combine yeast, 1 teaspoon sugar, and 1/4 cup warm water; stir well, and let stand 5 minutes or until bubbly.
Combine egg, remaining sugar, softened butter, potatoes, and salt in a large mixing bowl; beat at medium speed of electric mixer 5 minutes or until smooth. Stir in yeast mixture. Add 2 cups warm water alternately with enough flour to make a soft dough, beginning and ending with flour.
Turn dough out onto a lightly floured surface, and knead 10 minutes or until smooth and elastic. Place in a greased bowl, turning to grease top. Cover and let rise in a warm place (85°), free from drafts, 1 hour or until doubled in bulk.
Punch dough down; shape dough into 3 1/2-inch balls, and place 2 inches apart on greased baking sheets. Cover and repeat rising procedure 1 hour or until doubled in bulk.
Using a sharp knife, cut a shallow "M" on the top of each bun. Bake at 400° for 20 minutes or until golden brown. Brush tops with melted butter.