Whisk warm milk, yeast, and 1 tablespoon of the sugar together and let it stand until foamy.
Mix in the remaining sugar, half of the melted butter, eggs, flour, and salt until a soft dough forms.
Knead until smooth, place in a greased bowl, cover, and let rise until doubled, about 60 to 90 minutes.
Roll the dough into a rectangle, brush with the remaining butter, and scatter over brown sugar and cinnamon.
Roll up from the long side, cut into 12 pieces, and arrange in a greased baking dish. Let rise 30 minutes.
Bake at 350°F until puffed and lightly golden, about 22 to 25 minutes.
Beat cream cheese, powdered sugar, and vanilla until smooth, then spread over the warm rolls.