Combine sugar and cinnamon together in a bowl.
Roll the dough out evenly. It should be about an ⅛-1/4" thick and the long end should be about 16-18".
Give the room temperature brown butter a little mix to combine as it will be separated.
Spread brown butter across the dough evenly.
Sprinkle the cinnamon sugar mixture evenly over top of the butter.
Use your fingers to crumble the brown sugar evenly over the top of the rest of the filling.
Starting on the long end, roll up the dough.
Cut into 12 even rolls. The easiest way for me to get the rolls even is to cut the log in half, then each half in half again. Then each of those I cut into 3 even pieces.
Place evenly in a greased 9x13" pan. I like to spread them out a bit and make sure they aren't tightly rolled. I find this enables the insides of the rolls to puff and expand more.
Cover and allow to rise for 30 minutes. 10 minutes into the rise, preheat the oven to 350° F.
Once they are done rising, bake for 17-22 minutes until done. The tops and edges will just be starting to turn golden brown.
Make Frosting and Frost the Cinnamon Rolls
Beat the cream cheese and room temperature brown butter together on medium speed until completely smooth and lump free, about 1-2 minutes.
Add the vanilla and brown sugar and beat to combine.
Add the powdered sugar and mix until well combined and smooth.
Allow brown butter cinnamon rolls to cool about 10-20 minutes before frosting. This way they will still be warm and the icing will seep into the rolls, but they will be cool enough so that they won't get soggy. For frosting that sits on top of the rolls, wait until they have cooled completely. I double this frosting recipe for plenty of that nutty brown butter goodness.