Combine the water and milk in a microwave safe dish, and heat in the microwave in 20 second intervals until it reaches around 95 degrees (warm but not hot).
Add the yeast and sugar and stir. Leave for 5-10 minutes at room temperature, or until the yeast looks puffy.
In the bowl of a stand mixer fitted with a dough hook, add the bread flour and salt and mix until combined.
Separate an egg, adding the yolk to the mixer bowl. Discard or save the white.
Add an additional whole egg to the bowl.
Add the milk, water, yeast and sugar mixture, and mix until combined.
in 3 additions, add the softened butter. Mix until completely incorporated between each addition.
Knead until smooth, just a couple of minutes.
Cover the bowl and let the dough rise for 1 hour at room temperature, or until doubled in size.
After the first rise, cut the dough into 10 equal pieces. Weigh each piece if you want them to be really even.
Shape each piece into a ball (see video in post).
Place on a baking sheet lined with parchment paper, about 2 inches apart. Let the buns rise for 45 minutes.
Preheat oven to 375 degrees.
Mix an egg with 1 tablespoon of water in a small bowl. With a pastry brush, brush the egg wash on top of each bun.
After the egg wash, sprinkle sesame seeds on top of each hamburger bun.
Bake for 15-20 minutes, turning the tray halfway through.
Right after they come out of the oven, brush the top of each bun with melted butter. This will make them soft and delicious.
Let cool completely and enjoy!
