Place warm water in a small bowl; sprinkle with yeast, and let stand until foamy, about 5 minutes.
In a large bowl, whisk together warm 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 Fahrenheit, 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.
