Warm the whole milk to 110°F (43°C), stir in the sugar until dissolved, then whisk in the cooled melted butter and room-temperature egg until smooth.
In a stand mixer bowl, whisk together the flour, instant yeast, and salt, then add the wet milk mixture and mix on low until a shaggy dough forms, about 1 to 2 minutes.
Knead on low to medium-low speed for about 6 minutes until smooth, elastic, and slightly tacky, or knead by hand for 8 to 10 minutes.
Shape the dough into a ball, place it in a lightly greased bowl, cover, and let it rise in a warm spot for 1 hour to 1 hour 15 minutes until doubled in size.
Grease the 9x13-inch pan, punch down the dough, divide it into 15 equal pieces (about 55g each), shape each into a smooth ball, and arrange them in the pan so they will touch as they rise.
Cover the pan loosely and let the rolls rise for 30 to 45 minutes until puffy and touching, then preheat the oven to 375°F (190°C).
Brush the tops lightly with melted butter and bake at 375°F (190°C) for 18 to 22 minutes until deep golden brown and the centers reach about 190°F (88°C).
Immediately brush the hot rolls with the remaining melted butter, cool in the pan for about 5 minutes, then transfer to a wire rack or serve warm.