In the bowl of a stand mixer, whisk together the warm milk, yeast, and 1 tablespoon of the sugar. Let the mixture sit for 5 to 10 minutes until it becomes foamy.
Add the remaining sugar, 4 tablespoons of melted butter, and the egg to the yeast mixture. Whisk until well combined.
Attach the dough hook to the mixer. Add 2 cups of the flour and the salt. Mix on low speed until a shaggy dough forms.
Gradually add the remaining 1.5 cups of flour. Increase the speed to medium-low and knead for 5 to 7 minutes. The dough should be smooth, elastic, and pull away from the sides of the bowl while remaining slightly tacky.
Transfer the dough to a lightly greased bowl. Cover with a damp cloth or plastic wrap and let rise in a warm, draft-free spot for 60 to 90 minutes, or until doubled in size.
Gently punch down the risen dough to release air. Turn it out onto a lightly floured surface and divide into 12 equal-sized pieces (about 70-75g each).
Shape each piece into a tight, smooth ball by tucking the edges under. Place the rolls into a greased 9x13 inch baking pan in 3 rows of 4.
Cover the pan and let the rolls rise for a second time for 30 to 45 minutes until they look puffy and are touching each other.
While the rolls rise, preheat your oven to 350°F (177°C).
Bake the rolls for 18 to 22 minutes, or until the tops are golden brown and the internal temperature reaches 190°F (88°C) on an instant-read thermometer.
Remove from the oven and immediately brush the tops with the remaining 2 tablespoons of melted butter. Allow to cool slightly before serving.