Stir the warm milk, yeast, and 1 tablespoon of the sugar together. Rest for 5 minutes, until lightly foamy.
Whisk in the remaining sugar, melted butter, eggs, and salt. Add the flour gradually and mix until a soft dough forms.
Knead on a lightly floured surface for 6 to 8 minutes, until smooth and elastic. Place in a greased bowl, cover, and let rise in a warm spot for about 60 minutes, until doubled.
Roll the dough into a 16-by-12-inch rectangle. Spread with softened butter, then sprinkle evenly with brown sugar and cinnamon.
Roll the dough tightly from the long side into a log. Cut into 12 even pieces and arrange in a greased 9-by-13-inch pan, cut sides up. Cover and let rise for 30 minutes.
Heat the oven to 350°F. Bake for 20 to 22 minutes, until the tops are lightly golden and the centers are set.
Beat the cream cheese and butter until smooth. Mix in the powdered sugar, vanilla, and milk, then spread the icing over the warm rolls.
Notes
The dough should feel soft and slightly tacky, not wet. Let the rolls cool for a few minutes before icing so the frosting stays creamy.