1. Combine Wet Ingredients and Starter: In a large mixing bowl, whisk together the active sourdough starter, warm milk, granulated sugar, melted and cooled butter, and the egg until well combined. The mixture should look slightly frothy from the starter.

2. Add Dry Ingredients: Gradually add the salt and 3 1/2 cups of all-purpose flour to the wet mixture, mixing with a wooden spoon or a stand mixer with a dough hook until a shaggy dough forms. If the dough is too sticky, add the remaining 1/2 cup of flour, a little at a time, until it comes together and pulls away from the sides of the bowl.

3. Knead the Dough: Turn the dough out onto a lightly floured surface and knead for 5-7 minutes until it’s smooth and elastic. Alternatively, use a stand mixer with a dough hook on medium speed for 5-7 minutes. The dough should feel soft and spring back when gently poked.

4. First Rise (Bulk Fermentation): Lightly grease a clean bowl with a little oil. Place the kneaded dough in the bowl, turning it once to coat. Cover the bowl with plastic wrap or a damp kitchen towel and let it rise in a warm place for 3-4 hours, or until doubled in size. The dough should look puffy and have visible air bubbles.

5. Shape the Rolls: Gently punch down the risen dough to release the air. Turn it out onto a lightly floured surface and divide it into 12-15 equal pieces. Roll each piece into a smooth, round ball. Arrange the dough balls snugly in a greased 9×13-inch baking dish, leaving a little space between them for expansion.

6. Second Rise (Proofing): Cover the baking dish loosely with plastic wrap or a damp towel and let the rolls rise again in a warm place for 1-2 hours, or until they are puffy and nearly touching each other. They should look noticeably larger and softer.

7. Prepare for Baking: Preheat your oven to 375 F (190 C) during the last 15 minutes of the second rise. Just before baking, brush the tops of the rolls with the 2 tablespoons of melted butter. This will give them a beautiful golden crust.

8. Bake the Rolls: Bake for 20-25 minutes, or until the rolls are golden brown on top and sound hollow when tapped on the bottom. Remove from the oven and brush again with a little more melted butter for extra shine and flavor, if desired. Let them cool slightly before serving.