In a stand mixer bowl, whisk warm milk, sugar, and yeast. Let sit for 5-10 minutes until foamy. Add melted butter, egg, and sourdough discard and mix. Add flour and salt, then knead with a dough hook for 7-8 minutes until smooth and elastic.

Place dough in a lightly oiled bowl, cover, and let rise in a warm place for 60-90 minutes, or until doubled in size.

Combine softened butter, brown sugar, and cinnamon for the filling. Roll the risen dough into a 12x18 inch rectangle. Spread the filling evenly, leaving a ½-inch border on one long edge. Roll tightly into a log and pinch to seal.

Cut the log into 12 rolls and place in a greased 9x13 inch baking dish. Cover and let rise for 30-45 minutes until puffy.

Preheat oven to 375°F (190°C). Bake the rolls for 25-30 minutes, until golden-brown and the filling is bubbly.

While rolls bake, beat cream cheese and butter until smooth. Add powdered sugar, vanilla, and 1 tbsp milk; beat until creamy. Add more milk if needed for desired consistency. Let rolls cool for 5-10 minutes, then spread frosting generously over the top.