In the bowl of a stand mixer, combine the warm water, sugar, and yeast. Mix well, then let sit for 5-10 minutes, or until the yeast is frothy and bubbly.
Add the oil, eggs, salt, and half of the wheat flour into the yeast mixture and mix on low using the dough hook until combined - about 1-2 minutes.
Continue adding one cup of flour at a time, then mixing on low speed, until the dough pulls away from the sides and looks smooth. Once you have added in all of the flour that you need, knead the dough on medium speed for about 3 minutes. The dough should be smooth and elastic.
Cover the bowl with plastic wrap and let rise in a warm place until doubled in size, about 45-60 minutes.
Grease a 9" x 13" baking dish with butter or cooking oil.
Once your dough has doubled, transfer to a lightly floured surface and divide into 1 ½" dough balls (should about yield 20-24). Spray your hands with cooking oil and roll the balls until smooth. Place the shaped dough balls into the prepared baking pan so that they are about 1” apart or just barely touching.
In a small bowl, add the egg and water and whisk. Brush a thin, even layer over the tops of each roll using a pastry brush, then let rise again until doubled in size and touching, about 30-45 minutes.
Preheat your oven to 350 degrees F. Once your rolls have doubled in size, bake for 20-25 minutes, or until golden brown (an instant-read thermometer should read 190-195 degrees F when inserted into the center of your rolls).
When the dinner rolls are done, remove them from the oven and brush with melted butter. Serve warm. Enjoy!