1. In the bowl of a stand mixer fitted with the dough hook, combine the warm milk, warm water, instant yeast, sugar, and beaten egg.
2. Add the bread flour and all-purpose flour to the bowl, and mix until incorporated. Mix in the salt and butter. Knead on medium-low speed for about 10 minutes, or until the dough comes together into a soft yet tacky dough. Avoid adding too much extra flour.
3. Transfer the dough to a lightly oiled bowl. Cover with plastic wrap and let rise in a warm place until doubled in size, about 1 to 2 hours.
4. Line two baking sheets with parchment paper. Divide the dough into 8 equal parts. Gently roll each portion of dough into a ball and place four to a baking sheet, 2 to 3 inches apart. Cover with a clean towel and let rise again until puffy and almost doubled, 1 to 1 ½ hours.
5. Preheat the oven to 400°F. Brush the tops of the buns lightly with the egg wash. Sprinkle with sesame seeds if desired. Bake the buns for about 15 minutes, rotating halfway through baking, until the tops are golden brown.
6. Transfer to a rack to cool completely. Serve immediately or store in an airtight container in the freezer for up to one month.