To a glass measuring cup, add 1 cup water and sugar, and then stir to dissolve the sugar.
To the water sugar mix, add yeast and stir. Set the mixture aside to allow the yeast to proof (foam). This will take approximately 10 minutes or more.
Cook's Note: If you are using instant yeast instead of active dry yeast, you do not need to proof the yeast and can immediately move on to the next step.
Once the yeast has proofed, add the melted butter and 1 egg to the yeast mixture. Stir well until the butter and egg are fully incorporated.
To either a large bowl or the bowl of an electric mixer, add yeast mixture, 1/2 cup of flour, salt, and mix well.
Begin adding flour, one cup at a time, mixing well between each addition until a sticky dough mixture begins to form and begins to pull away from the sides of the bowl. (See video.)
Cook's Note: If you are mixing this by hand, the handle of a wooden spoon or a Danish dough whisk can be helpful. If you are using an electric mixer, use the dough whisk attachment.
Mixing by Hand: Once the sticky dough mixture has formed, it's time to knead it into a smooth dough. If you are doing this by hand, transfer the dough to a floured surface and knead the dough for about 10 minutes. If the dough sticks to the surface, add a little more flour to the surface and continue to knead the dough. A plastic bench scrape can also be helpful for this purpose.
Mixing by Hand: Once the dough is smooth, form it into a ball and make sure that any loose ends are tucked underneath the bottom of the ball.
Mixing with an Electric Mixer: If you are using an electric mixer, knead the dough at low speed for about 5 minutes. If you find that some of the sticky dough is sticking to the sides of the bowl, stop your mixer and scrape down the sides. If the sticky dough continues to stick the sides of the bowl and does not form into a ball, completely pulling away from the sides of the bowl, sprinkle in one tablespoon of flour at a time until the dough forms into a smooth mass.
Mixing with an Electric Mixer: Transfer the dough onto a floured surface and form the dough into a smooth ball, tucking in the loose ends underneath the bottom of the ball.
Once you have the dough formed into a ball, drizzle the olive oil in whatever bowl you are using and place the dough into the bowl. Turn the dough over and over several few times in the bowl to completely coat the dough with the oil.
Cover the bowl with plastic wrap, aluminum foil, or a tea towel, and let the dough rise in a warm place until it has doubled in size. This process will take about 1-2 hours. (The shorter rise time will be when using instant yeast. The longer rise time will be for when using active dry yeast.)
Once the dough has doubled in size, transfer the dough to a floured surface and pat it down to form it into a 5"x10" rectangle about 1/2 inch thick. Dust the dough lightly on top if it is exceptionally sticky.
Cut the rectangle of dough into eight equal pieces.
Form each piece of dough into a round ball, tucking the loose ends underneath the ball. Then pat the dough balls between your hands until they resemble a flat disk about 1/2 inch in thickness. (See video.)
Line a baking sheet with parchment paper or a silicone baking sheet and place each disk of dough onto the baking sheet about 1 inch apart.
Dust disks very lightly with flour and cover with a clean piece of plastic wrap, aluminum foil, or a tea towel.
Allow disks to rise in a warm place until doubled in size and look like hamburger buns. This will take approximately 30 minutes to 1 hour depending on what type of yeast you used.
While the disks are rising, preheat the oven to 375°F (190°C).
Whisk the remaining egg with the milk in a small bowl until completely combined. Set aside.
Once the disks have risen and look like hamburger buns, using a pastry brush, gently brush the tops of the buns with the egg/milk mixture. This is known as a "wash."
Immediately after brushing the wash onto the top of the buns, sprinkle them with the sesame seeds, if using.
Bake hamburger buns in the preheated oven until lightly golden brown on top. This can take between 15-20 minutes.
Remove baking sheet from oven and place on a heatproof surface. Allow the buns to cool slightly, then transfer to a wire cooling rack to finish cooling completely.
Once cooled, slice hamburger buns in half crosswise and serve.
Well wrapped, hamburger buns will stay fresh in the refrigerator for 5-7 days or 3 months in the freezer.