In a large bowl, whisk together 3 cups flour, sugar, salt, and yeast. In a separate bowl, combine warm milk, melted butter, egg, and discard. Add wet ingredients to dry, stir to form a shaggy dough, and knead on a floured surface for 5 to 6 minutes, adding remaining flour as needed until smooth and slightly tacky.

Shape dough into a ball, place in a greased bowl, cover, and refrigerate for 8 to 24 hours until puffy and nearly doubled.

On a floured surface, roll the cold dough into a 12x18-inch rectangle. Spread very soft butter over the dough, leaving a 1/2-inch border along one long side. Sprinkle evenly with brown sugar and cinnamon.

Roll the dough tightly from the long side opposite the border into a log. Pinch the seam, then cut into 12 equal slices. Arrange cut-side up in a greased 9x13-inch dish.

Cover and let rise in a warm spot until puffy, 45 to 60 minutes. Preheat oven to 350°F (180°C). Bake for 22 to 25 minutes until golden and set in the center.

Beat cream cheese and butter until smooth. Sift in powdered sugar, add vanilla and salt, and thin with milk as needed. Let rolls cool 5 minutes, then spread frosting over the warm rolls and serve.