Combine milk, water, yeast, and sugar in small bowl or 2-cup glass measure. Let stand 5 minutes until foamy.
Combine flour and salt in bowl of stand mixer fitted with dough hook attachment (or whisk together in large bowl if mixing by hand). Add water/milk mixture and mix on medium-low speed until combined. With mixer running, add two eggs, one at a time, waiting for eggs to become fully incorporated between additions. Add butter one tablespoon at a time until fully incorporated.
Continue to mix on medium-low speed for 2-3 minutes. If dough appears too sticky, add flour, 1 tablespoon at a time, until it pulls away from sides of mixer (it will still stick to bottom). Continue kneading until dough is smooth and elastic, 5-6 minutes longer.
Cover bowl with plastic wrap and let rise until doubled in volume (about 2 hours at room temperature, or 8 hours in the refrigerator). Turn dough out onto lightly floured surface, and using bench scraper or sharp knife, cut into 8 equal pieces approximately 4 ounces each.
Line two baking sheets with parchment paper. Form each dough chunk into a ball and place on baking sheets, four per sheet, about 3 inches apart. Spray with non-stick cooking spray (or brush with oil), cover loosely with plastic wrap, and allow to rise at room temperature until approximately doubled in volume (about two hours).
Adjust oven racks to lower-middle and upper-middle positions and preheat oven to 400°F. Gently press dough balls down until approximately 4 inches wide and 2 inches high. Beat remaining egg with one tablespoon water and brush entire exposed surface of bun. Bake for 7 minutes. Remove buns from oven, brush with remaining egg wash, and bake until deep, shiny, golden brown, rotating pans top to bottom and front to back half way through baking, 10-15 minutes longer. Transfer buns to rack and allow to cool completely.