In a large bowl, stir 2 cups of the flour, the sugar, 1/4 cup butter, salt and yeast with a wooden spoon until well mixed. In a 1-quart saucepan, heat the water and milk over medium heat, stirring frequently, until very warm and an instant-read thermometer reads 120°F to 130°F. Add the water mixture and egg to flour mixture. Beat with an electric mixer on low speed 1 minute, stopping frequently to scrape batter from side and bottom of bowl with a rubber spatula, until flour mixture is moistened. Beat on medium speed 1 minute, stopping frequently to scrape bowl. With a wooden spoon, stir in enough of the remaining flour, about 1/2 cup at a time, until dough is soft, leaves side of bowl and is easy to handle (dough maybe slightly sticky).