In a 4-cup measuring cup, with a spout if you have it, combine the warmed milk, melted butter, and the egg. Whisk until blended.

In the bowl of a stand mixer fitted with the dough hook, combine the flour, sugar, yeast and salt. Turn the mixer on low and slowly pour in the milk mixture and blend until a dough forms, about 2 minutes. Increase the speed to medium-low and mix until the dough is smooth and elastic, about 8 minutes. After 4 minutes of mixing you can add the additional flour if needed, one tablespoon at a time, just until the dough cleans the sides of the bowl but still sticks to the bottom. You may not need this additional flour depending on humidity, moisture in the butter, etc.

Line two large baking sheets with parchment paper and set aside. Melt 4 tablespoons of butter and have a pastry brush on hand.

Place the risen dough on a lightly floured work surface and divide into two equal parts. Place half the dough back in the bowl and cover with a clean dish towel. Working with the remaining half, pull and stretch the dough into a smooth and even 12-inch log. Cut the dough into 12 equal size pieces. Working with one dough ball at a time, roll into a smooth, tight ball using the palm of your hand. Cover the dough balls with the lightly greased plastic wrap while forming the rolls.

Use a small rolling pin or your hands to flatten each dough ball into an oval (about 4-inches long) with a depressed middle (higher on the sides). Lightly brush the center oval with melted butter, then fold in half to form the traditional Parker House roll shape. Press lightly on the edges to seal then transfer to a prepared baking sheet. The edges will not seal properly if buttered so try and keep the butter in the center of the oval. Repeat with the remaining dough balls spacing the rolls about 2-inches apart on the baking sheet. Cover the formed rolls with a lightly sprayed piece of plastic wrap and set aside in a warm location.