In a large mixing bowl, combine the warm milk, sugar, and yeast. Let it sit for about 5 minutes until it becomes frothy.
Add the melted butter, egg, and salt to the yeast mixture, and mix well until combined.
Gradually add the flour, one cup at a time, mixing with a wooden spoon or spatula until a soft dough forms.
Turn the dough out onto a floured clean surface and knead for about 5-7 minutes until smooth and elastic.
Place the dough into a greased bowl, cover with a kitchen towel, and let it rise in a warm area for about 1 hour, or until it doubles in size.
Punch down the risen dough to release the air and incorporate the shredded cheese evenly.
Divide the dough into 12 equal pieces, shape each piece into a ball, and place them into a greased roll pan or baking sheet.
Cover the rolls with the kitchen towel and let them rise for another 20-30 minutes until they puff up.
Preheat the oven to 375°F (190°C) and bake the dinner rolls for 15-20 minutes or until golden brown on top.
Remove the rolls from the oven and let them cool slightly before serving.