Stir the warm milk, yeast, and 1 tablespoon granulated sugar together and let stand until foamy, about 5 to 10 minutes.
Mix in the remaining granulated sugar, melted butter, and egg.
Add the flour and salt, then knead until the dough is smooth and slightly tacky, about 6 to 8 minutes.
Place the dough in a greased bowl, cover, and let rise until doubled, about 60 to 90 minutes.
Roll the dough into a 12-by-18-inch rectangle and spread it with the softened butter.
Mix the brown sugar and cinnamon, sprinkle it evenly over the dough, roll up from a long edge, and cut into 12 rolls.
Arrange the rolls in a greased 9-by-13-inch pan, cover tightly, and refrigerate for 8 to 16 hours.
Let the pan stand at room temperature for 45 minutes while the oven heats to 350°F.
Bake until puffed and golden, about 22 to 25 minutes.
Beat the cream cheese, confectioners sugar, vanilla, and enough milk for a spreadable icing, then spread it over the warm rolls.