In a large bowl, whisk together flour, yeast, sugar, and salt.

In a small saucepan, heat butter and milk until butter has just melted. Remove from heat to cool slightly.

In a medium bowl, whisk together warm water and one egg. Whisk in warm milk and butter mixture until combined. Pour over flour mixture and stir with a wooden spoon or rubber spatula until no dry flour remains and you have a shaggy, sticky dough. Tightly cover bowl with a lid or plastic wrap and let stand in a warm place for 2 to 4 hours until dough is bubbly and has doubled in size. Or tightly cover and chill for 12 to 24 hours to rise slowly in the refrigerator (My preferred way, for the best flavor and texture - see recipe note #4).

Line a baking sheet with parchment paper.

Turn risen dough out onto a lightly floured work surface and with lightly floured hands, first gather dough into a ball, then gently pat into a rectangle. With a dough scraper, divide dough into 8 equal pieces (use a kitchen scale for perfectly even buns). Shape each piece into a ball by first tucking edges under then pinching together to seal. Place pinched edge side down on work surface and gently roll with your palm to form a ball. Place 2 to 3 inches apart on prepared baking sheet. Cover loosely with a clean kitchen towel and let rise until buns are puffy and light to the touch, about 1 hour or up to 2 hours if dough is straight from the refrigerator.

When ready to bake, preheat oven to 400˚F.

Whisk together remaining egg with 1 tablespoon water and brush over buns with a pastry brush. Bake 15 minutes or until tops are golden brown, rotating sheet halfway through baking time. Transfer buns to a wire rack to cool completely.