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

2. Add Dry Ingredients and Mix: Add the all-purpose flour and fine sea salt to the wet ingredients. Mix with a wooden spoon or a stand mixer with a dough hook on low speed until a shaggy dough forms and no dry streaks of flour remain. Do not overmix at this stage.

3. Incorporate Butter: Gradually add the softened butter pieces to the dough, one piece at a time, mixing until each piece is fully incorporated before adding the next. This process might take 10-15 minutes in a stand mixer on medium-low speed. The dough will look messy at first but will eventually come together, becoming smooth, elastic, and pulling away from the sides of the bowl.

4. First Rise (Bulk Fermentation): Cover the bowl with plastic wrap or a damp kitchen towel and let the dough rise at room temperature (around 70-75 F / 21-24 C) for 8-12 hours, or until it has nearly doubled in size and looks bubbly. Alternatively, you can place it in the refrigerator for a longer, slower rise (12-24 hours), which can deepen the flavor.

5. Shape and Chill: Once the dough has completed its first rise, gently deflate it and turn it out onto a lightly floured surface. Shape the dough into your desired brioche form (a simple loaf or individual rolls). For a loaf, gently shape it into a log and place it into a lightly greased 9×5 inch loaf pan. Cover the pan loosely with plastic wrap and refrigerate for at least 2 hours, or preferably overnight. This chilling step makes the dough easier to handle and improves its texture.

6. Second Rise (Proofing): Remove the chilled dough from the refrigerator and let it proof at room temperature for 2-4 hours, or until it has visibly increased in size and looks puffy. It should pass the “poke test” – when gently poked with a floured finger, the indentation should spring back slowly. Preheat your oven to 375 F (190 C) during the last 30 minutes of proofing.

7. Egg Wash and Bake: Just before baking, gently brush the top of the brioche with the whisked egg wash. This will give it a beautiful golden-brown crust. Bake for 30-35 minutes, or until the top is deeply golden brown and an instant-read thermometer inserted into the center reads 200-205 F (93-96 C).

8. Cooling: Remove the brioche from the oven and let it cool in the pan for 10-15 minutes before transferring it to a wire rack to cool completely. Slicing it too soon can result in a gummy texture.