1. Combine the Wet Ingredients: In a large mixing bowl, whisk together the active sourdough starter, warm milk, granulated sugar, one large egg, egg yolk, and vanilla extract (if using). Make sure the sugar is mostly dissolved.

2. Add Dry Ingredients: Gradually add the all-purpose flour and salt to the wet mixture. Mix with a wooden spoon or your hands until a shaggy dough forms. It will look a bit rough and sticky at this stage.

3. Initial Kneading and Butter Incorporation: Turn the dough out onto a lightly floured surface. Knead for about 5 minutes until it starts to come together. Then, begin adding the softened butter, one piece at a time, kneading each piece in fully before adding the next. This process might seem messy at first, but keep going!

4. Develop the Dough: Continue kneading the dough for another 10-15 minutes until it becomes smooth, elastic, and passes the “windowpane test” (you can stretch a small piece thin enough to see light through it without tearing). Place the dough in a lightly oiled bowl, cover it with plastic wrap or a damp towel, and let it rise in a warm place for 4-6 hours, or until doubled in size. Alternatively, you can refrigerate it overnight for a slower, more flavorful rise.

5. Shape the Buns: Gently punch down the risen dough to release the air. Turn it out onto a lightly floured surface and divide it into 12 equal pieces. Roll each piece into a smooth, tight ball. Arrange the shaped buns in a greased 9×13 inch baking pan, leaving a little space between them.

6. Second Rise and Preheat Oven: Cover the pan loosely with plastic wrap or a damp towel and let the buns proof in a warm place for 1-2 hours, or until they look puffy and have nearly doubled in size. Towards the end of the proofing time, preheat your oven to 375 F (190 C).

7. Egg Wash and Bake: Once the buns are fully proofed, gently brush the tops with the beaten egg wash. If desired, sprinkle with coarse sea salt or sesame seeds. Bake for 20-25 minutes, or until the tops are golden brown and the internal temperature reaches 200-205 F (93-96 C) when checked with an instant-read thermometer.

8. Cooling: Remove the pan from the oven and let the buns cool in the pan for 10 minutes before transferring them to a wire rack to cool completely.