Dissolve the yeast in the warm water in a very large bowl, then stir in the sugar and let stand for 5 minutes.
Stir in the egg, melted butter, and salt. Stir in as much of the flour as you can with a wooden spoon.
Transfer the dough to a stand mixer fitted with a dough hook and mix on low speed for 6 to 8 minutes.
Alternatively: You can turn the dough out onto a lightly floured surface and knead by hand until the dough is smooth and elastic.
Shape the dough into a ball and place back in the lightly greased large bowl, turning one to cover the surface of the dough with grease. Cover with a clean hand towel and let rise in a warm place until the dough doubles in size, about 45 to 60 minutes.
Punch the dough down and turn out onto a lightly floured surface, divide the dough in half and cover the dough back up with the cloth and let rest for 10 more minutes. Meanwhile, lightly grease two 9x13-inch pans.
For Larger Rolls: Use a 15x10-inch pan and do not divide the dough.
Divide each half of the dough into 12 pieces and shape the rolls by gently pulling each dough piece into a ball, tucking the edges under and arranging on the baking sheets. If you are making larger rolls, make 15 equally sized balls of dough.
Cover the pans of rolls with hand towels and let rise in a warm place until the rolls have doubled in size, about 30 minutes.
Preheat the oven to 350 degrees F and brush the tops of the rolls with melted butter. Bake for 15 to 18 minutes or until the tops are golden brown. Brush the rolls with more butter after removing from the oven. Best served within 24 hours.