Activate Yeast: In a small bowl, dissolve 1 tbsp of sugar and yeast in warm milk. Let it sit for 5-10 minutes until frothy.
Mix Dough: In your stand mixer bowl, combine the yeast mixture, remaining sugar, melted butter, egg, and salt. Gradually add flour until a soft, tacky dough forms.
Knead: Knead on medium speed for 5-7 minutes until the dough is smooth and elastic.
First Rise: Place dough in a greased bowl, cover with a warm towel, and let rise in a warm place for 1 hour or until doubled in size.
Prepare Filling: While dough rises, mix brown sugar and cinnamon in a small bowl.
Roll & Fill: Punch down the risen dough. Roll it out on a floured surface into a 12x18 inch rectangle. Brush with softened butter and sprinkle evenly with the cinnamon-sugar mixture.
Shape: Tightly roll the dough into a log starting from the long edge. Cut into 12 even rolls using a sharp knife or dental floss.
Second Rise: Place rolls into a greased baking pan. Cover and let rise for 30 minutes until puffy. Preheat oven to 350°F (175°C).
Bake: Bake for 20-25 minutes until golden brown.
Make Icing: Whisk cream cheese, butter, powdered sugar, vanilla, and milk until smooth.
Serve: Drizzle icing generously over warm rolls and serve immediately.