Whisk starter, milk, and sugar together. Add egg, melted butter, vanilla, and 3½ cups flour with salt, mixing to a shaggy dough, adding more flour as needed until tacky.

Knead 6 minutes on a floured surface until smooth. Place in an oiled bowl, cover, and let rise 2–3 hours at room temp or refrigerate 8–12 hours for an overnight rise.

Punch down dough, roll into a 16x12-inch rectangle on a floured surface. Spread with soft butter, leaving a ½-inch border on the far long side. Sprinkle with brown sugar and cinnamon, press gently.

Roll up tightly from the long edge, pinch seam, slice into 12 rounds with floss or a serrated knife. Place cut side up in a greased 9x13-inch pan.

Cover and let rise until puffy, 45–60 minutes. Preheat oven to 375°F (190°C) and bake 24–26 minutes until golden.

Beat cream cheese, 3 tbsp butter, powdered sugar, vanilla, and milk to a thick glaze. Let rolls cool 10 minutes, then spread glaze over the top.