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Add the ingredients to your bread machine in the order they are listed, starting with the milk and ending with the yeast.
Set the machine to the Dough setting, most machines take about 1 hour and 30 minutes to finish the cycle.
Once the dough cycle is complete, transfer the dough to a lightly floured work surface. Use a pastry scraper tool or knife to divide it into 10 equal portions. Shape them into round buns by folding the dough up underneath each ball and pinching the seam closed. Spread the rolls evenly on a baking pan lined with parchment paper.
Cover the shaped buns with plastic wrap and let rise for 1 hour and 30 minutes or until they have almost doubled in size. It helps to keep them on top of the stove while the oven preheats.
Preheat the oven to 400°F.
Whisk together the second egg and tablespoon of water to form an egg wash. Use a pastry brush to brush evenly over the top of the buns to add shine and color.
Bake the buns for 10 - 13 minutes or until the internal temperature reads 200°F. They will brown very quickly at the end, watch carefully so they don't burn.