In a small bowl, microwave the shortening for about 30 seconds on high until mostly melted. Set aside to cool.

In a large bowl or the bowl of a stand mixer, mix the warm water, yeast, and sugar. Let stand for about 5 minutes until the yeast begins to bubble and foam.

Add the egg, salt, and about 2 cups of the flour. Beat with a spoon or run the mixer with a dough hook to combine, then add the lukewarm shortening and beat a minute or two to combine. Continue adding the flour in one-cup increments, beating or kneading by hand or running the stand mixer on low after each addition to fully incorporate. Resist the urge to add more than 5 total cups of flour, though you may add up to 1/2 cup more if needed. The finished dough should be slack, or slightly loose, or the rolls will be tough and dry.

Spray a large bowl with cooking spray and scrape and pull the finished dough into it. Spray the top of the dough with cooking spray, cover the bowl with plastic wrap, and set it aside to rise for about an hour, or until it has doubled in size. 

Spray a 9 x 13-inch baking dish with cooking spray. Once the dough has risen, oil your hands and gently deflate it. Pull off a handful of dough, about the size of a ping pong ball, and form it into a roll. Place each formed roll into the prepared baking dish, spacing them closely. 

When all the rolls are formed and in the baking dish, spray or oil a piece of plastic wrap and drape it loosely over the top dish, not touching the rolls. Set the baking dish aside and allow the rolls to rise for at least a half-hour, or until doubled in size.

Toward the end of the rise time, preheat the oven to 350° F. When the rolls have finished rising, place the pan into the oven and bake about 30 minutes, or until tops are browned and rolls are starting to pull away from the sides of the pan. After removing the rolls from the oven, let them rest for a few minutes. While they’re still hot, rub the tops with a tablespoon of butter or margarine on a piece of waxed paper or aluminum foil.