In a large mixing bowl, combine the warm milk, active sourdough starter, sugar, 2 tablespoons of melted butter, and the egg. Whisk the mixture together until it is smooth and completely combined.
Stir in the salt and 4 cups of flour with a wooden spoon until a shaggy dough forms. Turn the dough out onto a lightly floured surface and knead for 8 to 10 minutes, gradually adding the remaining 1/2 cup of flour as needed, until the dough is smooth, elastic, and only slightly sticky.
Place the dough in a large bowl that has been lightly greased with olive oil. Turn the dough over once to coat it in the oil. Cover the bowl tightly with plastic wrap and let the dough rise in a warm place for 4 to 6 hours, or until it has nearly doubled in size.
Gently punch down the dough to release the air. Divide it into 12 equal pieces and shape each piece into a smooth, taut ball by pulling the edges underneath. Place the rolls seam-side down on a parchment-lined baking sheet, spacing them about 2 inches apart.
Cover the rolls loosely with a clean kitchen towel and let them rise again in a warm place for 1 to 2 hours, or until they are puffy and have noticeably expanded.
While the rolls proof, preheat your oven to 375°F. Bake the rolls for 20 to 25 minutes, or until they are deeply golden brown on top and sound hollow when tapped on the bottom.
Immediately after removing the rolls from the oven, brush the tops with the tablespoon of melted butter. Transfer the rolls to a wire rack to cool for at least 15 minutes before serving.