1. In a large mixing bowl, combine the warm milk, granulated sugar, and active dry yeast. Stir gently and let it sit for 5-10 minutes until the yeast is foamy. Then, add your active sourdough starter and the large egg, whisking until well combined.

2. Gradually add the salt and 2 cups of the all-purpose flour to the wet mixture. Mix with a wooden spoon or a dough whisk until a shaggy dough forms.

3. Turn the dough out onto a lightly floured surface. Begin kneading, gradually adding the remaining 1 cup of flour, a little at a time, until the dough is less sticky but still soft. Knead for about 5-7 minutes until the dough starts to become smooth and elastic.

4. Now, add the softened butter pieces, one at a time, kneading each piece fully into the dough before adding the next. Continue for another 8-10 minutes until the dough is smooth, elastic, and no longer sticky. It should pass the windowpane test.

5. Lightly grease a clean bowl with a little oil. Place the dough in the bowl, turning it once to coat. Cover the bowl with plastic wrap or a clean kitchen towel and let it rise in a warm place for 2-3 hours, or until doubled in size.

6. Gently punch down the risen dough to release the air. Turn it out onto a lightly floured surface. Divide the dough into 12 equal pieces (about 60-70 grams each). Roll each piece into a smooth, tight ball. Place the shaped buns on a parchment-lined baking sheet, leaving about 2 inches between them.

7. Cover the baking sheet loosely with plastic wrap or a clean kitchen towel. Let the buns rise again in a warm place for 30-45 minutes, or until they look noticeably puffy and soft.

8. Preheat your oven to 375 F (190 C). In a small bowl, whisk the remaining egg with a tablespoon of water to create an egg wash. Gently brush the tops of the risen buns with the egg wash. Bake for 18-22 minutes, or until the buns are golden brown on top and sound hollow when tapped on the bottom. Let them cool on a wire rack before serving.