Add the sourdough starter, milk, flour, eggs, salt, sugar, and vanilla 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, add the softened butter one tablespoon at a time, scraping down the bowl as needed.
Increase the speed to medium and knead for 7–10 minutes, until the dough pulls away from the sides of the bowl and clings to the dough hook. *Brioche dough is known for being a wet dough. You're going to want to add more flour, and I urge you not to!
Cover the bowl and let the dough rest at room temperature for 6-10 hours, or until risen about 30–50%.
Note: Brioche dough rises slowly. In a cooler home (around 68°F), this can take closer to 12 hours. Watch the dough, not the clock!