In a glass measuring cup, combine 1 cup warm water, the milk, yeast and sugar. Let stand until foamy, about 5 minutes.
Meanwhile, beat 1 egg.
In a large bowl, whisk flours with salt.
Add butter and rub into flour between your fingers, making crumbs.
Using a dough scraper, stir in yeast mixture and beaten egg until a dough forms.
Scrape dough onto clean, unfloured counter and knead, scooping dough up, slapping it on the counter and turning it, until smooth and elastic, 8 to 10 minutes.
Shape dough into a ball and return it to bowl. Cover the bowl with plastic wrap and let rise in a warm place until doubled in bulk, 1 to 2 hours.
Line a baking sheet with parchment paper.
Using dough scraper, divide the dough into 12 equal parts. Gently roll each into a ball and arrange 2 - 3 inches apart on a baking sheet.
Cover loosely with a clean kitchen towel and let buns rise in a warm place for 1 - 2 hours.
Set a large shallow pan of water on the oven floor.
Preheat oven to 400 degrees with rack in center.
Beat remaining egg with 1 tablespoon water and brush some on top of buns.
Bake, turning sheet halfway through baking until tops are golden brown, about 15 minutes. Transfer to a rack to cool completely