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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.
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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.
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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).
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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.
