In the bowl of a standing mixer, whisk together the milk, yeast, sugar, egg, butter, and salt. Then, using a stout spoon or spatula, stir in the flour one cup at a time. Attach the dough hook to the mixer, and knead at medium speed until the dough becomes smooth and elastic, and it clears the sides of the bowl -- 4-5 minutes.
Transfer the dough to an unfloured work surface. Knead by hand until the dough is no longer sticky -- about 2 minutes. Put the bowl in a large, lightly-greased bowl. Flip the dough to grease its other side. Cover the bowl with cling film, and let rest until the dough has doubled in volume -- about 90 minutes.
Lightly grease a 9x13-inch baking dish. Deflate the dough, and then cut it into 15 fairly uniform pieces. Form each piece into a roll, and place in the prepared baking dish. Brush the tops of the rolls with melted butter. Cover with cling film and let rest until the rolls have doubled in volume -- 30-45 minutes. Meanwhile, preheat the oven to 350°F.
Bake until the rolls color lightly -- 15-20 minutes. Then immediately brush the tops of the rolls with (yet more) melted butter. Enjoy while hot or warm with -- you guess it -- butter!