1. Sprinkle yeast over warm water in bowl. Add sugar and salt. Stir to dissolve completely.
2. Add egg, butter and 3 cups flour. Beat at high speed 2 minutes. Gradually add Parmesan, onion flakes and cayenne. Stir in 1 cup flour.
3. Turn dough out onto surface. Knead until smooth and elastic, working in flour as needed to prevent sticking. Place in large bowl. Brush top with melted butter. Cover with plastic wrap. Let rise in refrigerator until doubled in bulk, 2 hours.
4. Punch dough down. Refrigerate, 1 to 3 days; punch down each day. Bake rolls all at once, or one batch a day for 3 days.For dozen rolls:
5. Remove one-third of dough from refrigerator; let stand at room temperature 2 hours. On floured surface, roll into 12-inch rope; cut in 12 pieces. Shape each piece into ball; tuck underneath to make smooth top. Place in greased round 8- or 9-inch round cake pan. Cover with towel; let rise in warm place until doubled in bulk, 30 to 45 minutes.
6. Heat oven to 375 degrees F.
7. Bake 15 to 20 minutes until golden brown. Brush with melted butter if desired.