Pour the water into a large mixing bowl. Sprinkle the yeast over the top and let sit for 5 minutes, or until mixture becomes cloudy. Whisk briefly. Add the salt, sugar and vegetable oil. Whisk until the salt and sugar have dissolved, about 1 minute.
Using a wooden spoon, add the flour 1 cup at a time, stirring after each addition. Transfer the dough to a lightly floured work surface. Knead the dough for approximately 3 minutes, or until soft and pliable. Coat a clean, large bowl with vegetable oil and add the dough, coating it completely with the oil from the bowl. Cover the bowl and let it sit in a warm place for 1 hour, or until the dough has doubled in size.
Coat a 9×13 baking dish generously with cooking spray and set aside. Place the dough onto a lightly floured surface and lightly knead for 3-5 minutes, or until dough is soft and pliable. Divide dough into 15 equal pieces and roll each piece into a ball with your hands. Place the balls of dough into the prepared baking dish, making five rows of three. Cover the baking dish and let it sit in a warm place for 2 hours, or until the dough balls have doubled in size.
Preheat the oven to 400°F. Gently brush the tops of the dough with the beaten egg. Bake in the preheated oven for 20 minutes, or until rolls are golden-brown. Cut at the seams and serve immediately.