Add the active sourdough starter, milk, flour, eggs, salt, sugar, and vanilla extract to the bowl of a stand mixer fitted with the dough hook attachment.
Mix on low to medium speed until a shaggy dough forms.
With the mixer running, 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 speed to medium-high and knead for 7–10 minutes, until the dough is soft, smooth, and slightly sticky, but not overly sticky. The dough should pull away from the sides of the bowl while clinging to the dough hook.
Resist the urge to add extra flour unless absolutely necessary. Brioche dough should be soft and slightly sticky, and adding too much flour can result in dense, dry sticky buns.
Cover the bowl and let the dough rest at room temperature for 6–10 hours, or until it has risen by about 30–50%. Once the dough has risen, you can place it in the fridge for up to 24 hours to make the recipe fit your schedule; otherwise, carry on to the next steps!
Brioche dough can rise slowly. In a cooler home, around 68°F, the first rise may take closer to 10 hours. Watch the dough rather than the clock.