In the bowl of a stand mixer, add the warm water, yeast, and sugar and mix to combine. Let sit for about 5 minutes, or until the yeast mixture starts to get bubbly.
Add the oil, egg, and salt to the yeast mixture and mix, then add 2 cups of flour and begin to mix using the dough hook attachment. At this point, the dough will be very sticky.
Continue to add up to ⅔ cup flour to the dough, until the dough starts to pull away from the edges of the bowl as you mix. Once you have added all of the flour, mix on low speed for about 2 minutes to knead. Turn the mixer off, cover the bowl loosely with a kitchen towel or plastic wrap and let the dough rise until light and puffy, or about 20-30 minutes.
Meanwhile, preheat the oven to 350 degrees F and grease a 9" x 13" baking dish with butter or non-stick cooking oil.
Transfer the dough from the bowl onto a flat surface and divide into 15 equal pieces. Roll into balls and place into the prepared baking dish, spacing them out evenly.
Cover loosely again and let the dough sit in a warm place for 15-30 minutes minutes, or until the rolls are about double the original size. Once risen, remove the towel and brush the rolls lightly with milk.
Bake in your preheated oven for 12-15 minutes, or until golden brown on top and the internal temperature reads 190 degrees F.
Remove from the oven and let cool for 1-2 minutes, then brush the tops with melted butter. Enjoy!