Make the Dough: In a large mixing bowl or stand mixer, combine warm water, warm milk, yeast, sugar, egg, egg yolk, butter, 1 cup of flour and salt. Whisk until well combined. Add another cup of flour and mix until a shaggy dough forms.
Knead for several minutes, adding a little flour at a time as needed, until the dough comes together. The dough should be soft, slightly sticky but not stick to clean, dry fingertips.
First Rise: Cover and let rise for 1 to 1 ½ hours, until doubled in size.
Shape Buns: Divide dough into 8 equal pieces (around 80 grams each). Roll each into a smooth ball and place on a parchment-lined baking sheet. Use your palm to gently flatten the balls to 3–3 ½ inches wide.
Second Rise: Cover loosely and let rise for another 1 to 1 ½ hours, until puffed.
Preheat oven to 375°F.
Egg Wash & Topping: Whisk egg with water until well combined and brush a thin layer of egg wash over buns. Sprinkle with sesame seeds if using.
Bake hamburger buns for about 14-18 minutes, until golden brown. Move to a cooling rack. Optional, brush tops lightly with melted butter.
