In a large mixing bowl, combine 2 cups of flour, sugar, yeast, and salt. Mix well.
In a separate bowl, combine warm milk, melted butter, and the egg. Whisk until blended.
Gradually add the wet ingredients to the dry ingredients. Mix until combined.
Slowly incorporate the remaining flour, 1 cup at a time, using a stand mixer with a dough hook or a spoon until a soft dough forms.
Knead the dough for about 5-7 minutes until smooth and elastic. If the dough is too sticky, add additional flour, a little at a time.
Place the dough in a lightly greased bowl, cover with a kitchen towel or plastic wrap, and let it rise in a warm place until it doubles in size, about 1 hour.
Once risen, punch down the dough and turn it out onto a lightly floured surface. Divide the dough into 12 equal pieces and shape each piece into a ball.
Place the dough balls into a greased 9x13 inch baking pan, leaving a little space between each roll. Cover and let rise again until doubled, about 30-45 minutes.
Preheat the oven to 350°F (175°C). Bake the rolls for 20 minutes or until golden brown.
Remove from the oven and brush the tops with melted butter before serving.