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).
Sprinkle flour lightly on a countertop or large cutting board. Place dough on floured surface. Knead by folding dough toward you, then with the heels of your hands, pushing dough away from you with a short rocking motion. Move dough a quarter turn and repeat. Continue kneading about 5 minutes, sprinkling surface with more flour if dough starts to stick, until dough is smooth and springy. Spray a large bowl with the cooking spray. Place dough in bowl, turning dough to grease all sides. Cover bowl loosely with plastic wrap; let rise in a warm place about 1 hour or until dough has doubled in size. Dough is ready if an indentation remains when you press your fingertips about 1/2 inch into the dough.
Spray the bottom and sides of a 13x9-inch pan with the cooking spray. Gently push your fist into the dough to deflate it. Divide dough into 15 equal pieces. Shape each piece into a ball; place in pan. Brush with melted butter. Lightly spray sheet of plastic wrap with cooking spray; cover the pan loosely with the plastic wrap, sprayed side down. Let rise in a warm place about 30 minutes or until dough has doubled in size. Remove plastic wrap.
Move the oven rack to a low position so that top of the pan will be in the center of the oven. Heat the oven to 375°F Bake 12 to 15 minutes or until golden brown. Remove from pan to a cooling rack. For a softer crust, brush tops of rolls with room-temperature butter, using a pastry brush. Serve warm or cooled.