Feed your sourdough starter several hours before you begin so it is bubbly and active. When it has risen and looks airy with a pleasant tangy smell, it is ready to use for sourdough dinner rolls.
In a large mixing bowl add the warm milk. Stir in the active sourdough starter until it blends into the liquid and looks milky and slightly foamy. Add the sugar and salt and stir until they start to dissolve.
Add the softened butter to the bowl in small pieces. Add most of the bread flour, keeping a small amount aside. Stir with a wooden spoon or dough whisk until a rough shaggy dough forms and no liquid remains at the bottom of the bowl. The dough should feel soft and a little sticky.
Turn the dough out onto a lightly floured surface. Knead gently for about eight to ten minutes. Use a push and fold motion until the dough becomes smooth and elastic. Add only small sprinkles of flour if it sticks badly. The dough should end up soft and slightly tacky rather than dry.
Shape the dough into a smooth ball and place it in a lightly oiled bowl. Turn the dough once so the surface has a thin coat of oil. Cover the bowl with a kitchen towel or reusable cover. Let the dough rise at room temperature until it has grown by about half to three quarters in size. In a warm kitchen this may take three to five hours.
For deeper flavor and easier timing you can place the bowl in the refrigerator for eight to twelve hours after the dough has started to rise. When you are ready to shape the sourdough dinner rolls, remove the bowl and let the dough sit at room temperature for about thirty minutes so it softens slightly.
Grease a metal baking pan with a thin layer of butter or oil. Turn the risen dough onto a lightly floured surface and gently press it into a rectangle to release any very large air pockets. Use a bench scraper or sharp knife to cut the dough into twelve equal pieces.
To shape each roll, take one piece of dough and pull the edges toward the center to form a rough ball. Flip it seam side down. Cup your hand over the dough and move it in small circles on the surface to tighten the top so it becomes smooth and round. Place the shaped ball in the prepared pan. Repeat with the remaining pieces, arranging them in neat rows with a little space between each one.
Cover the pan loosely and let the sourdough dinner rolls rise at warm room temperature until they look puffy and almost touch each other. This may take about two to three hours. When ready, the rolls should jiggle slightly if you nudge the pan and a gentle press on one roll should leave a small mark that slowly springs back.
Near the end of the final rise, heat the oven to three hundred seventy five degrees Fahrenheit. When the oven is hot, lightly brush the tops of the rolls with a small amount of milk or melted butter if you like a softer and more golden crust. Place the pan on the middle rack.
Bake the sourdough dinner rolls for about twenty to twenty five minutes. The tops should be a warm golden color and the rolls should feel light. If the tops brown very quickly while the centers are still pale, lower the oven temperature a little or cover the pan loosely with foil for the last few minutes.
Remove the pan from the oven and set it on a cooling rack or heat safe surface. While the rolls are still warm, brush the tops with melted butter for a soft shiny finish. Sprinkle a small pinch of flaky salt or chopped herbs if you like. Let the rolls cool for about ten minutes, then serve warm as pull apart sourdough dinner rolls.