In a glass measuring cup with a spout, combine the water, yeast, and sugar. Stir and let rest for 2 minutes. The yeast should foam up; if it doesn't, your yeast is no longer good and you need to start over with new yeast.
In the bowl of a standing mixer fitted with the dough hook attachment, place the flour, butter, egg, and salt. Pour in the yeast mixture. Mix on low speed for two minutes, then increase the heat to medium-high and continue mixing until the dough is smooth and soft, about five minutes.
Place the dough in a bowl, cover, and let rise until doubled in bulk, about 30 minutes, depending upon your kitchen.
Line two baking sheets with parchment paper. With your fist, gently press down the dough in its center. Turn the dough out onto a clean surface and divide it into eight pieces.
Form each piece into a ball, pulling the edges towards each other into the center. Flip the dough over so the seam is touching the counter. Cup your hands around the ball and drag it gently towards you.
With your palm, press down until the dough is about half an inch thick, then release. Arrange four buns per sheet, placing them seam side down and leaving them plenty of room to rise.
Cover the rolls with lightly oiled plastic wrap and let rest until puffy and almost doubled, about 45 minutes.
Preheat your oven to 375°F. Brush the egg wash over the tops of the buns. If desired, sprinkle with sesame seeds.
Bake the Hamburger Buns on the lower rack for eight minutes, then rotate the pans and switch their positions. Continue baking until golden and cooked through, about seven to ten minutes more. Let cool, then slice in half horizontally.
