In the bowl of a stand mixer fitted with the dough hook, combine the active sourdough starter, warm milk, flour, eggs, salt, and sugar. Note: If your kitchen runs cold and you want this to ferment overnight, keep the milk cold.
Mix on low speed until a shaggy dough forms.
With the mixer running on low, add the softened butter one tablespoon at a time, allowing each addition to fully incorporate before adding more. Scrape down the sides of the bowl as needed.
Increase the mixer speed to medium-high and knead for 7–10 minutes, or until the dough becomes smooth, elastic, and begins pulling away from the sides of the bowl.
Tip: Brioche-style dough is naturally soft and slightly sticky. Resist the urge to add extra flour for the softest, fluffiest pockets.