In a large mixing bowl, whisk together the active sourdough starter, lukewarm milk, granulated sugar, melted butter, and eggs until smooth.
Add the salt and 520 g of flour to the bowl and mix with a spoon or in a stand mixer on low speed until a shaggy dough forms.
Knead by hand or with the dough hook for 7–10 minutes, adding small amounts of the remaining flour only if needed, until the dough is soft, slightly tacky, and elastic.
Shape the dough into a ball, place it in a lightly greased large bowl, cover tightly, and let it rise at room temperature for 6–8 hours or until roughly doubled and puffy.
In a medium bowl, combine the soft butter, brown sugar, cinnamon, and vanilla for the filling and mix into a smooth, spreadable paste.
Lightly flour your work surface, turn out the risen dough, and gently deflate it by pressing it into a rectangle with your hands.
Using a rolling pin, roll the dough into a large rectangle about 40 x 30 cm (16 x 12 inches), keeping the thickness even.
Spread the cinnamon filling evenly over the surface of the dough, leaving a 1 cm (½ inch) border along one long edge.
Starting from the opposite long edge, roll the dough up tightly into a log, pinching the seam to seal.
Trim the ends if desired, then cut the log into 12 equal slices using a sharp knife or dough scraper.
Arrange the rolls cut-side up in a greased 9x13-inch baking pan, spacing them evenly.
Cover the pan tightly with plastic wrap or a clean towel and let the rolls rise at room temperature for 2–4 hours, or refrigerate overnight for a slower rise.
When ready to bake, remove the pan from the refrigerator (if chilled) and let the rolls sit at room temperature until noticeably puffy, about 45–60 minutes.
Preheat the oven to 180°C (350°F) while the rolls finish proofing.
Bake the rolls for 22–28 minutes, until lightly golden on top and the centers read about 88–93°C (190–200°F) if checked with a thermometer.
While the rolls bake, make the icing by whisking together the cream cheese and butter until smooth, then adding powdered sugar, vanilla, salt, and just enough milk or cream to reach a thick, spreadable consistency.
Let the baked rolls cool in the pan for 10–15 minutes, then spread or drizzle the cream cheese icing over the warm rolls.
Serve the cinnamon rolls warm and store any leftovers covered at room temperature for 1 day or refrigerated for up to 3 days.