Start by proofing your yeast. In a large mixing bowl or the bowl of your stand mixer, add warm water. Sprinkle the yeast and sugar on top and allow to sit for 5 minutes. The mixture should start to bubble.
Add in the egg and salt. Stir to combine.
Add the dough hook to your mixer. Start mixing in your flour 1 cup at a time, until you reach 3 cups. Then switch to adding 1/2 cup at a time, until the dough starts to pull away from the sides. This should take somewhere between 3 1/2 - 4 1/2 cups total depending on the flour you're using.
Let the mixer run for 3-4 minutes, or if kneading by hand, knead until the dough is smooth and elastic.
Transfer to a large greased mixing bowl. Cover with a clean towel and place in a warm spot to rise for about an hour.
At the hour mark, or once dough has doubled in size, you can start making your rolls. To do this, dump your dough onto a floured surface and shape into a rectangle. Cut the rectangle in half with a dough cutter. Divide each half into 6-8 even sized pieces. Shape into a ball and place in a greased 9X13 baking dish. If you're not bothered by slightly uneven rolls, you can also pinch off pieces of dough - a little larger than a 1/4 cup scoop - and shape into dough balls that way. Allow to rise for an additional 20 minutes, or until doubled in size, in a warm place. Brush the tops of the rolls with an egg wash before baking for a golden brown crust.
Bake in a 375 degree oven for 20-25 minutes.