Dough: In a large bowl, combine 1 cup flour and yeast. Heat milk, sugar, butter, and salt in a saucepan until warm. Add to flour mixture along with egg. Beat until smooth. Stir in enough remaining flour to form a soft dough. Turn onto a floured surface and knead until smooth and elastic, about 6-8 minutes. Place in a greased bowl, cover, and let rise until doubled, about 1 hour.
Filling: Punch down dough. Roll out into a 12×9 inch rectangle. Spread melted butter over dough. Combine brown sugar and cinnamon, sprinkle over butter. Roll up jelly-roll style, starting from a long side. Cut into 12 slices.
Baking: Place rolls in a greased 9×13 inch baking pan. Cover and let rise until doubled, about 30 minutes. Bake at 375°F for 20-25 minutes or until golden brown.
Icing: In a small bowl, combine 1 cup powdered sugar, 1 tablespoon milk, and 1/2 teaspoon vanilla extract. Drizzle over warm cinnamon rolls.
