Pour the warm milk into a small bowl, sprinkle the yeast over the top, and let it rest for about 5 minutes until it becomes bubbly and active. In another large mixing bowl, combine the flour, sugar, and salt by whisking them together.
Mix in the melted butter, egg, and prepared yeast mixture, stirring until the ingredients form a shaggy dough. Transfer the dough to a lightly floured surface and knead for 8 to 10 minutes, working it until it feels soft and elastic.
Place the smooth dough in a greased bowl, cover it with a clean towel, and let it rise in a warm spot for about 1 hour, or until it doubles in size.
In a small bowl, combine the brown sugar and cinnamon until evenly mixed. Gently press down the risen dough, then roll it into a rectangle measuring roughly 16 by 12 inches. Spread the softened butter in an even layer over the surface, and scatter the cinnamon-sugar mixture across the top.
Starting from a long edge, roll the dough tightly into a log, keeping it even as you go. Cut the log into 12 even slices and place them in a greased 9×13-inch baking pan. Cover the dish and allow the rolls to rest for about 30 minutes, or until they look light and puffy.
Preheat your oven to 350°F (175°C). Bake the rolls for 18 to 20 minutes, until the tops turn golden and the centers are baked through.
As the rolls bake, beat the cream cheese and butter together until the mixture is smooth and creamy. Add the powdered sugar, vanilla, and milk, and mix until creamy.
After removing the rolls from the oven, allow them to cool for a few minutes, then spread the icing generously over the warm tops so it melts slightly into each roll. Serve warm for the ultimate treat.