Heat the milk just until feels warm (about 100 degrees)
Sprinkle the yeast on top and leave it sit for 10 minutes.
Whisk in the egg, butter and vanilla.
Using the whisk attachment on a stand mixer, combine the flour and sugar on low for 3 minutes.
Replace whisk with dough hook attachment.
Pour wet ingredient mixture in the center of dry ingredients and mix on low speed until dough comes together.
Increase speed to medium and knead until the dough sticks to the hook (6-8 minutes).
Remove from hook and shape dough into a ball.
Clean mixer bowl and lightly grease with butter.
Return the dough to the bowl and turn to coat.
Cover with towel and allow to rise until doubled (about 1 1/2 hours).
Flour your work surface, and roll the dough into roughly 12x15" rectangle.
With longer side facing you, spread butter over the the surface, except for 1/2" edge opposite you.
Mix together your brown sugar, sugar and cinnamon and sprinkle mixture evenly over buttered area.
Roll the dough towards the unbuttered edge, trying not to leave gaps.
Gently rub water along the unbuttered edge and continue rolling to seal up the roll.
Cut the roll into 12 even pieces.
Line a 9x13" pan with parchment paper and plae rolls in the pan, leaving a little space between each. You can use multiple smaller pans if desired.
Cover with towel and allow to rise about 45 minutes.
Preheat the oven to 325.
Bake until golden brown (20-30 minutes).
Cool for 15 minutes.
In a medium bowl, beat together cream cheese and butter until blended.
Add in powdered sugar and beat until incorporated.
Add in vanilla and beat until incorporated.
If too thick, add milk to thin out slightly.
Spread icing evenly over rolls.