Warm the milk to exactly 110°F using a kitchen thermometer, pour into a large mixing bowl, and sprinkle the yeast and sugar over the milk. Let sit for 5-10 minutes until foamy and bubbly; if it doesn't foam, start over with fresh yeast.
Add the egg and ¼ cup of the melted butter to the yeast mixture and whisk until well combined. Add the salt to the flour in a separate bowl, then gradually add the flour mixture to the wet ingredients, one cup at a time, stirring until a shaggy dough forms.
Turn the dough out onto a lightly floured surface and knead by hand for 8-10 minutes (or 5-6 minutes with a stand mixer dough hook), adding small amounts of flour if too sticky, until smooth, elastic, and it springs back when poked. Form into a ball.
Grease the large mixing bowl, place the dough ball in it, and turn to coat all sides. Cover tightly with a clean kitchen towel or plastic wrap and let rise in a warm, draft-free spot for 1-1.5 hours until doubled in size.
Punch down the risen dough to release air bubbles, turn out onto a lightly floured surface, and divide into 16 equal pieces (about 2 ounces each). Roll each piece into a smooth ball and place seam-side down in a greased 9x13 inch baking pan, spacing evenly so they touch.
Cover the pan with a kitchen towel and let the rolls rise for another 30-45 minutes until puffy and nearly doubled in size, touching each other and springing back slowly when gently pressed.
Preheat the oven to 375°F. Brush the risen rolls generously with 2 tablespoons of the remaining melted butter and bake for 15-18 minutes until golden brown and the internal temperature reaches 190°F.
Remove the pan from the oven and immediately brush the hot rolls with the remaining melted butter for extra flavor and shine. Let cool in the pan for 5 minutes, then serve warm.