In large bowl, mix 2 cups of the flour, the sugar, salt and yeast. In 1-quart saucepan, heat water, milk and 1/2 cup butter until very warm (120°F to 130°F). Beat warm liquid and egg into flour mixture with electric mixer on low speed until moistened, scraping bowl frequently. Beat on medium speed 3 minutes, scraping bowl frequently. With spoon, stir in 2 1/2 to 3 cups of the flour until dough pulls cleanly away from side of bowl.
On floured work surface, knead in remaining 1 1/4 to 1 3/4 cups flour until dough is smooth and elastic, 8 to 10 minutes. Grease large bowl with shortening or cooking spray. Place dough in bowl; cover loosely with greased plastic wrap and cloth towel. Let rise in warm place (80°F to 85°F) 45 to 60 minutes or until light and doubled in size.
Gently push fist into dough to deflate; divide in half. To make Pan Rolls, lightly grease 2 (13x9-inch) pans with shortening or cooking spray. Divide each half of dough into 16 equal pieces. Shape each into ball, pulling edges under to make smooth top. Place 16 balls, smooth side up, in each pan. Cover; let rise in warm place 20 to 30 minutes or until light and doubled in size.
Heat oven to 400°F. Uncover dough; bake 16 to 20 minutes or until golden brown. Immediately remove rolls from pans; place on wire racks. Cool 15 minutes. Brush with melted butter. Serve warm or cool.