In the bowl of a standing mixer, whisk together the water and yeast. Then whisk in the egg, butter, sugar, and salt. Use a stout spoon to stir in half of the flour. When the first addition of flour is moist, stir in the remaining flour.
Attach a dough hook to the standing mixer. Knead at low speed for 2 minutes. Then turn the machine off, and let the dough rest for 2 minutes. At "medium" speed, knead until the dough becomes smooth and elastic -- about 7 minutes.
Use damp fingers or a damp spatula to scrape the dough onto a non-floured work surface. Form the dough into a ball, and then place the ball, seam-side up, in the large greased bowl. Flip the dough to grease its other side. Cover the bowl with plastic wrap, and let rise in a warm location until doubled in volume -- 90 minutes to 2 hours.
With damp fingers, gently punch down the dough, and turn it onto a non-floured work surface. Pat the dough into a rectangle. Use a bench scraper or knife to cut the rectangle into 8 equal pieces. Form each piece into a ball (please watch video above!), and arrange the balls on the prepared baking sheet in a zig-zag pattern. Loosely cover with plastic wrap, and let rise until doubled in volume -- 30-45 minutes. Meanwhile, preheat the oven to 375°F.
Just before baking, brush the buns with the egg wash, and, if you are using them, sprinkle the tops with sesame seeds. Bake until the buns are richly bronzed -- 15-17 minutes. Cool to room temperature on a wire rack.