In a large bowl or stand mixer, combine the warm water, warm milk, sugar, and yeast. Let sit for 5 minutes until slightly foamy.
Add the egg, butter, flour, and salt. Mix to form a dough.
Knead for 10 minutes (8 minutes with mixer) until smooth and elastic.
Cover and let rise in a warm place until doubled in size, about 1 to 1 ½ hours.
Divide the dough into 8 equal portions (about 95-105 g each).
Shape each portion of dough into a tight ball.
Place the buns on a parchment-lined baking sheet, spaced apart. Gently flatten each bun slightly with your palm.
Cover loosely and let rise again for about 15 minutes, until noticeably puffy.
Preheat oven to 375°F (190°C).
Brush the tops with egg wash.
If using, sprinkle each bun with sesame seeds.
Bake for 15-18 minutes, or until the buns are golden and the internal temperature reaches 190°F (88°C).
Remove from the oven and optionally brush with melted butter while warm for a softer crust.
