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

2. Add Dry Ingredients: Gradually add the salt and 3 1/2 cups of all-purpose flour to the wet mixture. Use a wooden spoon or a stand mixer with a dough hook to mix until a shaggy dough forms.

3. Knead the Dough: Turn the dough out onto a lightly floured surface and knead for 5-7 minutes, or if using a stand mixer, knead for 3-5 minutes until the dough is smooth and elastic. It should be slightly tacky but not sticky.

4. First Rise (Overnight Cold Fermentation): Place the dough in a lightly oiled bowl, turning it once to coat. Cover the bowl tightly with plastic wrap or a damp kitchen towel and refrigerate for 8-12 hours, or up to 24 hours. The dough will slowly rise and develop flavor.

5. Shape the Rolls: The next day, remove the dough from the refrigerator. Gently punch it down and turn it out onto a lightly floured surface. Divide the dough into 12 equal pieces (about 2 ounces each). Shape each piece into a smooth, round ball.

6. Second Rise (Proofing): Arrange the shaped rolls in a greased 9×13 inch baking dish, leaving a little space between them. Cover the dish loosely with plastic wrap or a damp towel and let them rise in a warm place for 1-2 hours, or until they are visibly puffy and almost doubled in size. They should look soft and pillowy.

7. Bake the Rolls: Preheat your oven to 375 F (190 C). Once the rolls have proofed, remove the cover and bake for 18-22 minutes, or until they are golden brown on top and sound hollow when tapped on the bottom.

8. Finish with Butter: As soon as the rolls come out of the oven, brush them generously with the remaining 2 tablespoons of melted butter. This adds shine and a lovely richness. Serve warm.