In a skillet, melt the butter over medium-low heat. Stir in the milk and set aside.
In your stand mixer with the paddle attachment, mix starter, egg, and sugar. Pour in the butter-milk mixture and combine well.
Add flour and salt, mixing until no dry pockets remain. Switch to the dough hook and knead on low speed for 8–10 minutes, until smooth and stretchy (passes windowpane test).
Transfer dough to a greased bowl, cover, and let rise until doubled (6–10 hours depending on kitchen temperature).
Grease a 9×13-inch baking dish. Divide dough into 15 equal portions (about 60–70g each) and shape into rolls. Arrange in a 3×5 layout in the dish.
Cover and let rise again until puffy, about 1½–3 hours.
Preheat oven to 350°F (175°C). Bake rolls for 25–30 minutes until golden brown.
Brush tops with melted butter and sprinkle lightly with salt.