These buns are the perfect accompaniment to a juicy hamburger. You can choose the size you want, but I like to make 8-10. I find 10 seems like a good manageable size for adults and kids alike. 8 for all adults. You can also knead this by hand easily, but for speed, I prefer the mixer.
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 (use the bowl for your stand mixer and it will save you an extra bowl during clean up) stir together the flours (3 cups bread and 1/2 cup all-purpose) with salt. Add butter and rub into flour between your fingers, making crumbs. Put the dough hook into your stand mixer, and mix in the yeast mixture and beaten egg until a dough forms. As the dough comes together, you can you increase the speed a little until the dough begins to leave the sides of the bowl and slaps back against it. You may need to add a bit more flour at this point so the dough becomes stretchy, but be careful to add as little as possible, as that keeps the finished buns lighter. Keep this up for 8-10 minutes until the dough is smooth and elastic.
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 or a baking silpat. Divide dough into 8-10 equal parts. Gently roll each into a ball and arrange 2 to 3 inches apart on baking sheet. Cover loosely with a clean kitchen towel and let buns rise in a warm place for 1 to 2 hours.
Set a large shallow pan of water on 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, until the tops are golden brown, and the bottoms are very lightly colored, about 15 minutes. Transfer to a rack to cool completely. Slice and serve as is, or, to really amp it up, grill with butter before making your burgers.