In a small bowl or coffee mug, microwave the 1/2 cup water for 20-30 seconds, until it is lukewarm but not hot.
Add the yeast and 2 teaspoons sugar and stir. Let this mixture sit for 5 minutes. Watch it so it doesn't overflow!*
In a glass measuring cup, microwave 2 cups milk for 1-2 minutes until it is warm but not hot.
In a large bowl or stand mixer, add 1/3 cup sugar, kosher salt, and shortening, and 2 cups of the flour. Stir together with a wooden spoon.
Add the warm milk, yeast mixture, and 1 egg. Mix well and let rest 5 minutes. At this point I started using the dough hook on my mixer.
Add 4 cups of flour, one cup at a time, mixing in between. Use your best judgment to continue adding flour until the dough is smooth and pulling away from the sides of the bowl, up to 5 or even 6 cups. Don't add so much that the dough becomes stiff. It should be soft and rather sticky, but still workable. The less flour you use, the more tender your rolls will turn out. See notes.
Once all the flour has been incorporated, knead for 5-6 minutes, using your hands or the dough hook.
Grease a large bowl well with oil or butter. Scrape all the dough into the bowl, shape it together, then turn it over so that the top is greased.
Cover loosely with a tea towel or plastic wrap. Let rise until doubled, about 1 hour.
Punch down the dough. Divide the dough into 24 equal pieces. Shape each piece into a ball (pinch one end with your fingers).
Place the rolls in 2 well-greased pans. I used one 9x13 pan and one 2 quart casserole dish.
Cover the pans with tea towels and let them rise in a warm spot for another 30-45 minutes, until doubled in size. When they are almost done rising, preheat your oven to 375 degrees F.
Bake at 375 for 12-15 minutes or until golden brown all across the top.
Remove from the oven and brush with melted butter.
*If your yeast mixture is NOT getting foamy, then abandon ship! You killed your yeast with too hot water probably. Better to start over now than wind up with sad flat rolls in 2 hours.**My aunt's recipe says to use 4-5 cups of flour, and that's what I had originally published. Dough is finicky, and how much flour you need to add depends on a lot of different factors. But I consistently use at least 5 cups of flour when I make this, and sometimes up to 6 or even 6 and 1/2 cups, so I've changed the recipe to reflect that. You are going to have to use your best judgement here. It should be a very sticky dough, but you do want it to come together. Use the least amount of flour that you can in order to get it to form into a workable dough.