In a large bowl, combine 1-1/4 cups of the flour and the yeast. Set aside. In a small saucepan, heat and stir the milk, the 1/4 cup sugar, the 1/4 cup butter, and the salt just until warm (120 degrees F to 130 degrees F) and butter almost melts. Add milk mixture to flour mixture; add egg. Beat with an electric mixer on low to medium speed for 30 seconds, scraping sides of bowl constantly. Beat on high speed for 3 minutes. Using a wooden spoon, stir in as much of the remaining flour as you can.
Turn dough out onto a floured surface. Knead in enough of the remaining flour to make a moderately soft dough that is smooth and elastic (3 to 5 minutes total). Shape dough into a ball. Place dough in a lightly greased bowl; turn once to grease the surface. Cover and let rise in a warm place until double in size (about 1-1/2 hours).
Grease baking sheet; set aside. Punch down dough. Turn out onto a lightly floured surface. Cover and let rest for 10 minutes. Roll out to a 20x10-inch rectangle. Brush with the 2 tablespoons melted butter.
In a small bowl, combine the 1/2 cup sugar, the chopped walnuts, and cinnamon; sprinkle evenly over dough. Starting from a long side, roll up dough into a spiral. Evenly cut into eight slices.
Being careful not to cut all the way through, snip or cut each slice into quarters. Place slices, seam sides down, 3 inches apart on the prepared baking sheet; spread each slightly to form a fan. Cover; let rise in a warm place until almost double in size (30 to 45 minutes). Preheat oven to 350 degrees F. If desired, brush rolls with water and sprinkle with additional sugar.
Bake rolls in the preheated oven for 12 to 15 minutes or until golden. Serve warm. Makes 8 rolls.