Stir together the water, sugar, yeast, and salt in the bowl of a stand mixer. Let sit until the mixture looks frothy, about 5 minutes. If your mixture does not look frothy your yeast is probably inactive and you should start again with new yeast.
Once the yeast mixture is frothy, stir in oil and egg. Add 3 cups of flour, and using the paddle attachment, mix on low for 1 minute, until mixture is well combined. Add the rest of the flour and mix for another minute, until dough begins to pull away from the sides of the bowl.
Transfer the dough to another lightly greased bowl (this will help the dough to rise and not cling to any leftover scraps of dough in the mixing bowl), cover with a towel, and set in a warm spot for 1 hour, until the dough doubles in size.
Punch down the dough and weigh out to 2-ounce portions. (If you don’t have a scale, eyeball 16 portions of equal size). To shape a roll, use your thumb to make an indention in the center, then fold the dough inward until a ball forms. Repeat with the remaining dough portions.
Brush a sheet pan with melted butter, place rolls on the pan, and cover with a towel. Set in a warm spot for 30 minutes, until the rolls double in size, while you heat the oven to 400°F.
After the rolls finish their second proof, brush tops of rolls with melted butter and bake for 7 minutes, rotate, then continue baking for another 7 minutes. When ready, the rolls should be golden brown.