Stir the warm milk, yeast and 1 tablespoon sugar together in a large bowl, then wait 5 to 8 minutes until a foamy cap forms.
Whisk in the remaining sugar, egg, 3 tablespoons melted butter and salt until evenly combined.
Add 3 cups flour and stir into a shaggy dough, then knead on a lightly floured surface for 7 to 9 minutes until smooth, elastic and slightly tacky.
Work in up to 1/4 cup remaining flour only if the dough sticks heavily to your hands; keep it softer than sandwich-bread dough.
Place the dough in a lightly buttered bowl, cover and let it rise in a warm spot for 60 to 75 minutes until doubled.
Press out the gas, roll the dough into a 12-by-8-inch rectangle and cut it into 12 equal rectangles.
Arrange the pieces barely touching in a buttered 9-by-13-inch pan, cover and rise for 30 to 40 minutes until puffy and joined at the edges.
Bake at 350°F for 16 to 18 minutes until the tops are evenly golden and the center rolls sound hollow when tapped.
Brush the hot rolls with the remaining 2 tablespoons melted butter and cool for 10 minutes before pulling them apart.
