Add the active sourdough starter, warm milk, flour, eggs, salt, sugar, and vanilla extract to the bowl of a stand mixer fitted with the dough hook.
Mix on low to medium speed until a shaggy dough forms.
With the mixer running on low speed, add the softened butter one tablespoon at a time, allowing each addition to incorporate before adding the next. Scrape down the sides of the bowl as needed.
Increase the mixer to medium speed and knead for 7–10 minutes, or until the dough pulls away from the sides of the bowl and clings to the dough hook.
Note: Brioche dough is naturally very soft and sticky. Resist the urge to add additional flour, even if it seems too wet. The dough will become smoother and easier to work with after fermentation and chilling.
Transfer the dough to a lightly greased bowl. Cover and let rise at room temperature for 6–10 hours, or until it has increased in volume by about 30–50%.
Tip: Brioche dough ferments slowly. In a cooler home (around 68°F/20°C), fermentation may take closer to 10 hours. Watch the dough rather than the clock.
Refrigerate the dough for at least 2 hours or overnight. Chilled dough is much easier to roll and shape.