1 stand mixer with a paddle and a dough hook a 3 inch round cookie cutter.
Bring milk to a bare simmer in a small saucepan over medium heat. Remove from heat and allow to cool to 105 to 115°F.
Meanwhile, stir together warm water, yeast, and 1/2 tsp sugar in mixer bowl until yeast has dissolved. Let stand until foamy, about 5 minutes.
(If mixture doesn't foam, start over with new yeast).
Add butter, warm milk, and remaining 1/4 cup sugar to yeast mixture and mix with paddle at low speed until butter has melted, then mix in eggs until combined well.
Add salt and four cups of the flour and mix, scraping down bowl as necessary, until flour is incorporated. Beat at medium speed for 1 minute.
Switch to dough hook and beat in remaining 2 cups flour at medium speed until dough pulls away from the sides of the bowl, about 2 minutes; if necessary add more flour 1 Tablespoon at a time. Beat five minutes more (dough will be sticky.).
Transfer dough to a large lightly oiled bowl and turn to coat. Cover bowl tightly with plastic wrap and let rise in a warm draft free place until doubled, about 2 1/2 hours.
Butter 2 large baking sheets. Punch down dough, then roll out on a lightly floured surface with a floured rolling pin into a 14 inch round (about 1/2 inch thick). Cut out as many rounds as you can with a floured cutter and arrange 3 inches apart on the baking sheets.
Gather scraps, re-roll, and then cut out more rounds.
Loosely cover buns with oiled plastic wrap and allow to rise in a draft free place until they hold a finger mark when gently poked, 1 1/2 to 2 hours.
Preheat oven to 375°F with racks in upper and lower thirds.
Mix egg and water for egg wash.
Brush buns with egg wash and bake, switching pans halfway through baking, until tops are golden and the bottoms are golden brown and they sound hollow when tapped, 14 to 20 minutes.
Buns can be frozen, wrapped well, for about a month.
Cooks Notes: If you don't have a stand mixer, stir ingredients together in same sequence with a wooden spoon until a dough forms. Knead dough on a floured surface, incorporating just enough flour to keep dough from sticking, until dough is smooth and elastic, 7 to 8 minutes.