In a 4-cup measuring cup, combine the warmed milk, melted butter, and egg. Whisk until blended. The mixture should be warm, not hot, to protect the yeast.
In a bowl of a stand mixer fitted with the dough hook, combine the flour, sugar, yeast, and salt. Mix on low. Gradually pour in the milk mixture, blending until a dough forms, about 2 minutes.
Increase the speed to medium-low and mix until the dough is smooth and elastic, approximately 8 minutes. After 4 minutes, you can add additional flour if needed, one tablespoon at a time, until the dough cleans the sides of the bowl but still sticks to the bottom.
Scoop the dough out onto a lightly floured work surface. Knead the dough by hand until it forms a smooth, round ball. Place the dough in a large, lightly oiled bowl, cover it with plastic wrap, and put it in a warm area to rise until doubled in size, about 1 to 1½ hours.
While the dough is rising, line two large baking sheets with parchment paper. Melt 4 tablespoons of butter in preparation for later.
Once the dough has risen, place it back on a lightly floured surface and divide it into two equal portions. Cover one portion with a clean dish towel.
Work with the remaining dough half by stretching it into a 12-inch log and cutting it into 12 equal pieces. Roll each piece into a smooth ball using your palm and cover with lightly greased plastic wrap while forming the rolls.
Flatten each dough ball into an oval shape (about 4 inches long) with a depressed middle, higher on the sides. Brush the center of each oval with melted butter, then fold them in half to create the signature Parker House roll shape. Press the edges lightly to seal.
Transfer the shaped rolls to the prepared baking sheet, spacing them about 2 inches apart. Cover with lightly sprayed plastic wrap and let them rise in a warm location until doubled in size, about 45 minutes to 1¼ hours.
Preheat the oven to 350 degrees, adjusting the racks to the upper-middle and lower-middle positions. Once the rolls have risen, lightly brush them with melted butter, then bake until golden brown, about 20 to 25 minutes. Be sure to rotate the pans halfway through baking.
Once baked, let the rolls cool slightly and serve warm, brushed with additional melted butter if desired. Enjoy the delicious taste of homemade Parker House Rolls!