In a small bowl, combine the warm water, warmed milk, yeast, and sugar. Let stand until bubbly, about 5 minutes.
In a large bowl, whisk together the bread flour, all-purpose flour, and salt. Add the butter. Using your fingers, rub the butter into the flour mixture, making crumbs. Add the yeast mixture and the beaten egg to the bowl. Using a dough scraper or silicone spatula, stir until a moist dough forms.
Scrape the dough onto an unfloured counter. At this point, the dough should be very moist and sticky, not really able to be kneaded traditionally. You will be tempted to dust with flour but resist. Instead of traditionally kneading, use your dough scraper or silicone spatula to scoop up the dough and then slam it back on the counter. Turn and repeat. Just keep going. After about 5 to 7 minutes, your dough should be still moist but not quite as sticky.
Shape dough into bowl and place into a greased bowl. Cover with plastic wrap and let rise until doubled, 1 to 2 hours.
Line a large baking sheet with parchment paper and set aside.
Using a dough scraper, divide the dough into 8 equal pieces. (I like to weight the dough ball and then divide that weight by 8, to get a weight for each piece. I then weigh out each piece to that weight, to get 8 exactly equal pieces.)
If your dough is still too sticky at this point to form into balls, very lightly dust with a tiny bit of four.
Form each of the eight pieces of dough into balls. Place balls on parchment-lined baking sheet about 2 to 3 inches apart. Cover loosely with a clean kitchen towel and let rise until doubled again, 1 to 2 hours.
Set a large shallow roasting pan of water on the lowest rack of the oven. Preheat oven to 400° F / 204° C, with a rack in the center of the oven, as well.
When buns have doubled, beat the remaining egg with 1 tablespoon water and brush the buns thoroughly. Bake, turning sheet 180 degrees halfway through baking until the tops are golden brown, about 15 minutes.
Immediately transfer buns to a rack to cool completely.