In a stand mixer bowl, combine active sourdough starter, milk, melted butter, sugar, and egg on low. In a separate bowl, whisk flour and salt. Gradually add dry ingredients to wet, then knead on medium for 8-10 minutes until dough is smooth and pulls away from the bowl sides.

Transfer dough to an oiled bowl, cover, and let rise in a warm place for 4-6 hours, or until it has increased in size by about 50%.

Turn risen dough out onto a floured surface and roll into a 12x18 inch rectangle. Spread softened butter over the dough. Mix brown sugar and cinnamon, and sprinkle it over the butter.

Roll the dough into a tight log and cut into 12 rolls. Place them in a greased 9x13 inch baking dish.

Cover the dish tightly with plastic wrap and refrigerate for 8-18 hours for the overnight cold proof.

Remove rolls from the refrigerator and let them proof at room temperature for 1-2 hours until slightly puffy. Preheat oven to 375°F (190°C).

Bake for 25-30 minutes until golden brown. Let cool for 10-15 minutes.

While rolls bake, beat cream cheese and butter until smooth. Add powdered sugar and vanilla, then add milk until desired consistency is reached.

Spread frosting over the warm rolls and serve immediately.