Make the Dough: In a large bowl, mix warm milk and yeast. Whisk in the egg, sugar, melted butter, and salt. Gradually add flour, stirring until the dough is tacky but not sticky. Knead for 3 minutes until smooth. Let rise in a greased bowl for 1–1 ½ hours in a warm place, until doubled in size.
Roll Out the Dough: On a floured surface, roll the dough into a 17 × 10-inch rectangle. Spread softened butter over the surface, then sprinkle evenly with brown sugar and cinnamon mixture.
Shape the Rolls: Starting at one long edge, tightly roll the dough into a log. Pinch the seam to seal, then cut into 12 equal pieces. Place rolls in a greased 9 × 13-inch pan.
Second Rise: Cover with plastic wrap and let rise for 45–60 minutes until visibly puffed.
Bake the Rolls: Preheat the oven to 350°F. Brush rolls with melted butter and bake for 20–22 minutes, until golden and centers register 200°F. Let cool slightly.
Make the Frosting: Beat butter and cream cheese until smooth. Add vanilla, confectioners’ sugar, and salt, beating until creamy. Adjust sugar to taste.
Frost and Serve: Generously spread frosting over warm rolls. Serve immediately.