Place water in a small bowl; sprinkle with yeast, and let stand until foamy, about 5 minutes.
In a large bowl, whisk together milk, butter, sugar, salt, and 2 eggs. Whisk in yeast mixture. (Save the remaining egg for the egg wash.)
Using a wooden spoon, stir in 6 cups flour, 1 cup at a time, until you have a soft, shaggy dough (if necessary, add up to ½ cup more flour).
Turn dough out onto a floured work surface; knead until smooth and elastic, 5 to 10 minutes.
Butter the inside of a large bowl; place dough in bowl, turning to coat. Cover bowl with plastic wrap; let stand in a warm spot until dough has doubled in size, about 1 ¼ hours.
Butter two 13-by-9-inch baking pans. Divide dough in half. Roll each half into a 15-inch rope; cut each rope into 15 1-inch pieces. Press each piece into a disk, then shape into a ball.
Arrange dough balls in prepared pans. Cover pans loosely with plastic; let stand in a warm spot until rolls have doubled in size, about 1 ¼ hours (2 hours more if frozen). Preheat oven to 375°F, with racks in upper and lower thirds. In a small bowl, beat remaining egg until blended; brush onto rolls.
Bake until golden brown, about 20 minutes, rotating pans back to front and top to bottom halfway through. Let rolls cool 15 minutes before serving.
