Brown the butter by placing in a medium saucepan over medium high heat and melt. Leave to continue cooking, stirring occasionally, until brown nutty flecks appear. At this point, transfer the butter immediately to a bowl so it doesn't burn, making sure to scrape up those brown flecks- that's where all the flavor is! Cool before using.
Warm milk to around 110F (43C), do this by putting in microwave safe bowl or cup and microwaving gently for 30-45 seconds. It should be like warm bath water.
Transfer warm milk to the bowl of an electric mixer and sprinkle yeast on top. Add in sugar, eggs and browned butter. Mix until well combined. Next, stir in flour and salt with a wooden spoon until a dough begins to form.
Place a dough hook on a stand mixer and knead dough on medium speed for 8 minutes. Dough should form a nice ball and be slightly sticky. If you need, to sprinkle in another tablespoon or two of flour, but don't add too much.
Transfer dough ball to a lightly oiled bowl, cover with plastic wrap and allow dough to rise in a warm place for 1 hour to 1 ½ hours, or until doubled in size.
After dough has doubled in size, transfer dough to a well-floured surface and roll out into a 14x9 inch rectangle. Spread softened butter over dough, leaving a small margin at the edges of the dough. I like using a silicone brush for this.
Mix together the brown sugar and the cinnamon for the filling in a small bowl.
Sprinkle the brown sugar cinnamon mixture evenly over the butter. Pat it into the butter by pressing down with your fingertips.
Tightly roll dough up, starting from the 9-inch side and place seam side down making sure to seal the edges of the dough as best you can. Trim the very edges of the log off, then cut the rest of the dough log into 9 equal 1 inch thick pieces.
Lightly grease a 9x9 inch baking pan and line with parchment paper. Place the rolls in the pan, touching each other. Cover with plastic wrap and a kitchen towel and let rise again for 30-45 minutes. Meanwhile, preheat the oven to 350F (180C).
Remove plastic wrap and towel and bake cinnamon rolls for 23-35 minutes or until just slightly golden brown on the edges. Don't overbake, so that they stay soft and gooey in the center, you want the edges to be a light golden brown.
While baking the rolls, make the cream cheese frosting. Beat together the softened cream cheese, softened butter, vanilla and salt until creamy and combined. Reduce speed to low and add powdered sugar. Mix until well combined. Store in the fridge until ready to use.
After the cinnamon rolls are out of the oven, let them cool for 5-10 minutes, before spreading frosting evenly over the type of the pan of rolls. You'll have extra frosting, feel free to serve that on the side! Enjoy warm and gooey!