In a large bowl or the bowl of a stand mixer, stir together the milk, water, yeast, and sugar. Let it sit for five minutes, or until bubbly and foamy.
Add the salt, one egg, butter, and two cups of flour, and mix until combined. Add additional flour, ⅓ cup at a time, until the dough pulls away from the sides of the bowl and comes together into a ball. The dough should be slightly sticky to the touch, but should roll into a ball if rolled between your thumb and finger.
Move the dough to a greased bowl, cover, and let rise one hour, or until doubled.
After the dough has risen, gently deflate it and divide it into nine equal portions. Roll each piece of dough into a ball, then flatten the ball slightly with your hand. Place on a baking sheet lined with silicone or parchment. Repeat with the remaining dough. Cover and let rise for one hour, or until doubled.
While the dough is rising, add a pan full of water to the bottom rack of the oven.
Preheat the oven to 400°F. Whisk the remaining egg with 1 teaspoon of water, then brush the egg mixture over the buns.
Bake 15 minutes, or until golden brown.