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1
Combine milk, water, yeast, and sugar in small bowl or 2-cup glass measure. Let stand 5 minutes until foamy.
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2
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.
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3
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.
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4
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.
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5
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).
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6
Adjust oven racks to lower-middle and upper-middle positions and preheat oven to 400u00b0F. 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.
