Heat the oven to its lowest setting for 2 minutes, then turn it OFF. Bring a kettle of water to a boil and set it aside — you will use both for the rise.
Whisk the warm milk, yeast, and sugar together in a large bowl and let it sit for 5 minutes until foamy. Whisk in the melted butter, egg, and salt.
Add the flour and mix until a shaggy dough forms, then knead for 5 minutes by hand or 3 minutes with a dough hook, until smooth and slightly tacky.
Cover the bowl. Pour the boiled water into a pan on the oven's bottom rack, put the dough on the middle rack, and close the door. Let it rise 25 minutes, until nearly doubled.
Roll the dough on a floured surface into a 16x12-inch rectangle. Spread the very soft butter edge to edge, then mix the brown sugar and cinnamon and scatter it evenly over the butter.
Roll the dough up tightly from the long edge. Cut into 12 rolls using unflavored dental floss slid underneath and pulled through — a knife flattens the spiral.
Arrange the rolls in a greased 9x13-inch pan. Return them to the warm oven (with the water pan) for a 15-minute second rise, until puffy and touching.
Take the rolls and the water pan out. Heat the oven to 350°F, then bake for 20 to 24 minutes, until golden and just set in the center.
Beat the cream cheese, butter, powdered sugar, vanilla, and milk until smooth. Let the rolls cool 10 minutes, then spread the frosting over them while still warm.