Add the starter, milk, flour, eggs, salt, sugar, and vanilla to the bowl of a stand mixer fitted with the dough hook.
Mix on low–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 speed to medium and knead for 7–10 minutes, or until the dough pulls away from the sides and clings to the dough hook.
Cover and let rise in a warm place until nearly doubled, 2–5 hours, depending on room temperature. (Optional: perform 1–2 stretch and folds in the first hour to speed fermentation.)
Once risen, transfer the covered dough to the refrigerator for a cold overnight ferment, 8–12 hours (up to 24 is fine).