Combine the ½ cup + 1 tbsp water and yeast in a small saucepan. Warm over medium heat until very thick and creamy. Remove from heat and let cool to lukewarm.
In a large bowl, combine the remaining 1 cup milk, the sugar, butter, and salt. Cool until warm (120°F to 130°F) and add the eggs. Beat until smooth.
Add the yeast mixture and 3 cups of the flour. Beat until a soft dough forms. Knead in the remaining flour to make a slightly tacky dough that is smooth and elastic (5 to 5 minutes). Place in a lightly greased bowl, turning to grease the top. Cover; let rise in a warm place (80°F to 85°F) for 1 hour.
Punch dough down. Let rest 10 minutes. Grease a 13x9-inch baking pan. Divide dough into 20 equal pieces. Shape each into a ball. Arrange in prepared pan. Cover and let rise in a warm place for 30 to 45 minutes.
Preheat oven to 350°F. Whisk together the remaining egg and 1 tbsp water. Brush rolls lightly with egg wash. Bake 25 minutes or until golden brown. Cool slightly. Makes 20 rolls.
