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.
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 1/2 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 1/4 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. (To make ahead: Wrap pans well, and freeze, up to 2 months.) Cover pans loosely with plastic; let stand in a warm spot until rolls have doubled in size, about 1 1/4 hours (2 hours more if frozen).
Preheat oven to 375 degrees, 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.
The rolls can be shaped in advance and frozen for up to 2 months. No need to defrost -- just add 2 hours to the second rise.