To brown the butter, place it in a large heavy bottomed saucepan over medium heat. Let it melt, and then stir constantly until it starts to bubble and deepen in color to an amber color and small brown flecks appear at the bottom of the saucepan. Remove from heat immediately and pour out into another large bowl to avoid burning. Get all those brown flecks out, that's where the flavor is. Let it cool slightly while you prepare the rest of the ingredients.
Warm the milk in the microwave, between 30 seconds to 1 minute or until it is lukewarm. You can also warm it on the stove.e It shouldn't be too hot to the touch that it's uncomfortable, but should feel like bath water.
Add the yeast to the warmed milk, and stir to dissolve as best as you can.
In the bowl with the brown butter, add the sugar and eggs and stir until combined with a wooden spoon. Add in the milk and yeast mixture and mix until combined.
Add the flour and salt, and mix again until combined and no flour patches are apparent. If it's easier, use your hands and knead a little to combine, just until you get a cohesive dough.
Cover with a damp kitchen cloth and rest in a warm place for 1 hour, or until dough has doubled.
After dough has rested and grown in size, lightly flour a surface, and dump dough onto it. Cover again with kitchen towel and let it rest for another 10 minutes.
While dough is resting, prepare your filling. Mix the brown sugar and cinnamon in a bowl, then add the softened butter and keep mixing until it forms a paste, you can use your fingertips.
Roll the dough out with a lightly floured rolling pin into a roughly 18 inch by 14 inch rectangle.
Sprinkle the filling evenly over the dough, leaving just a thin 1/2 inch border around the edges without filling so it doesn't overflow.
Roll up the cinnamon rolls into a long log, and using a sharp knife, slice evenly into 12 pieces.
Place into a baking pan that's been lightly greased and covered with parchment paper. It's fine if the cinnamon rolls touch. Rest this pan covered with a kitchen towel for another 30 minutes or so until puffy and nearly doubled. Preheat the oven to 350 F (180 C) so that it's ready to go after this rest.
Bake 20-25 min or until golden brown on the edges and paler in the center, and puffy. Let them stand 5-10 minutes before frosting.
While baking, make the frosting. Beat together the softened butter, softened cream cheese, vanilla and salt until creamy and combined. Reduce speed to low and add icing sugar. Mix until well combined.
Frost the warm rolls as generously as you like - you can save some frosting for the side if anyone wants extra. Enjoy warm and fresh out the oven!