Place warm water in a large mixing bowl. Sprinkle yeast over the warm water and set aside for 5 minutes to allow the yeast to dissolve. Meanwhile, oil two baking sheets or line with parchment paper and set aside.
Add egg, egg whites, oil, and entire contents of the bread mix to the yeast mixture. Using an electric mixer fitted with a paddle attachment, mix on low for 2 minutes. Turn the mixer off and scrape down the sides of the bowl with a rubber spatula then continue to mix on medium speed for 6 minutes. If preparing by hand, mix with a wooden spoon until the ingredients are very well combined.
For buns, portion dough into 1/3 cup (75g) portions; for rolls, portion dough into 3 Tbsp (50g) portions. With wet hands, gently shape each piece into a ball and place on the prepared baking sheets. Lightly brush or spray each piece with oil. Cover with plastic wrap and let rise in a very warm place until doubled in size, about 40 minutes. While the dough rises, preheat the oven to 350°F.
When the dough has doubled in size, remove plastic wrap, (if necessary, reshape any misshapen pieces with wet hands) and transfer sheet trays to the preheated oven. Bake until golden, rotating trays halfway through baking, about 15 – 20 minutes for rolls and 20 – 25 minutes for buns. Let cool at least 5 minutes before serving.
