Pour the warm water into a large mixing bowl and stir in the sugar and active dry yeast. The water should be approximately 110°F. Allow the mixture to stand for 2–5 minutes, or until a noticeable layer of foam develops and the yeast becomes fragrant.
While the yeast activates, thoroughly grease the bottom and sides of a 9 × 13-inch baking pan with oil or butter to prevent the rolls from adhering during baking.
Add the vegetable oil, salt, and room-temperature egg to the foamy yeast mixture. Stir until the wet ingredients are generally combined.
Gradually work the all-purpose flour into the liquid mixture, stirring after each addition. Continue until all the flour has been absorbed and a rough, shaggy dough develops.
Knead directly in the mixing bowl for approximately 4 minutes. Continue working the dough until it becomes smooth, soft, and slightly tacky without being excessively sticky.
Lightly coat your hands with vegetable oil or cooking spray. Divide the prepared dough into 15 evenly sized portions, then roll each piece between your palms to form a smooth, firmly shaped ball.
Position the dough balls seam-side down in the prepared 9 × 13-inch pan, arranging them in three rows of five. Leave approximately 1/2 inch of space between the rolls to provide room for expansion.
Loosely cover the pan with plastic wrap or a damp kitchen towel. Preheat the oven to 375°F. Once it reaches the proper temperature, switch the oven off.
Place the covered pan inside the warm, turned-off oven. Let the dough proof for 30–45 minutes, until the rolls appear noticeably puffy and have nearly doubled in volume. They should feel soft and should have expanded to occupy most of the pan.
Remove the risen rolls from the oven, then preheat the oven once again to 375°F.
Return the pan to the fully preheated oven and bake the rolls for 18–20 minutes. They are ready when the tops are richly golden, the upper surfaces feel set, and the sides where the rolls meet are lightly golden. If the oven tends to bake hot, begin checking them at the 18-minute mark.
Remove the baked rolls from the oven and leave them in the pan for 2–3 minutes. If desired, generously brush the warm tops with melted butter for additional richness and a glossy finish. Serve while still warm.