Activate yeast: In a medium bowl, whisk together milk, sugar, and yeast. Let stand until foamy, about 5 minutes.

Make dough: Place flour and salt in a stand mixer bowl fitted with a dough hook attachment. With mixer running on low speed, add milk mixture and egg to flour mixture.

Beat on low speed until combined and a shaggy dough forms, 2 to 3 minutes.

With mixer still running on low speed, gradually add 5 tablespoons of the melted butter, and beat until butter is completely incorporated, 3 to 4 minutes. (It takes some time for butter to incorporate into the dough.)

Increase mixer speed to medium, and beat until dough is smooth and elastic, 8 to 10 minutes.

Let dough double: Cover bowl with plastic wrap, and set aside at room temperature until doubled in size, 1 to 2 hours.

Divide dough: Turn dough out onto a clean work surface. Divide into 12 equal pieces (about 2 ounces each). Working with 1 piece of dough at a time, divide each into 3 equal pieces (about ¾ ounce each).

Fill muffin tray with dough balls: Roll each piece into a tight ball, and dip into 3 tablespoons of the melted butter. Place 3 balls together in each muffin well of an ungreased 12-cup muffin tray.

Let rolls rise: Loosely cover tray with plastic wrap; set aside, and let rise at room temperature until doubled in size and dough springs back slowly when gently pressed, 45 minutes to 1 hour. Meanwhile, preheat oven to 350°F.

Bake rolls: Bake in preheated oven until puffed and golden brown, 12 to 14 minutes.