Line a baking sheet with parchment paper or a silicone baking mat.
Combine the warm water, warm milk, yeast and sugar in the bowl of a stand mixer. Stir gently to combine. Let stand for 5 minutes.
Add the salt and egg to the bowl. Use the paddle attachment to combine on low speed.
Add both flours and mix on low speed until combined, then mix in the butter pieces.
Switch to the dough hook and knead for 8 minutes. The dough will be very soft and slightly sticky.
Transfer the dough to a floured work surface and knead a few times with your hands to form it into a ball. Cut the dough into 8 pieces. Shape each piece into a round ball and place on the prepared baking sheet.
Loosely cover the dough balls with a clean kitchen towel and let rest for 15 minutes.
Meanwhile, preheat oven to 200 degrees C (180 fan).
Combine the egg and 1 tablespoon milk in a small bowl, stirring with a fork. Remove the towel and brush the tops of the buns with the egg wash. Bake hamburger buns in the oven for 13-15 minutes, until tops are golden.