Make the dough: In the bowl of a stand mixer fitted with a dough hook attachment, combine the yeast and warm milk. Let it stand for 5 minutes until bubbly, then turn the mixer on low. Add 1/2 C of flour, then the sugar and browned butter, egg and salt. Increase the speed to medium and mix about 2 minutes.
One cup at a time, add the dough until it's fully incorporated and the dough pulls away from the sides of the bowl. The dough should be soft, but not stiff. Transfer the dough to a large greased bowl, cover with a kitchen towel or plastic wrap, and let rise in a warm place until doubled in size, about an hour.
Punch the dough down and flip it out on a lightly floured surface. Using a lightly floured rolling pin, roll the dough out into a 12x18 size rectangle.
Make the filling: Place butter in a saucepan and melt over medium heat. Stir with a whisk until foaming, and pull off the heat right when the butter smells nutty and starts to turn a golden brown. If you pull it off too late, it will burn, if you pull it off too soon, you won't get the full brown butter flavor. So watch it closely! Scrape brown butter and bits into a separate bowl to cool while preparing the other ingredients.
In a small bowl, combine the brown sugar and cinnamon. Spread as much as you can of the cooled melted butter over the flattened dough with a pastry brush. You might have to drizzle the rest on after adding the cinnamon sugar. Sprinkle the cinnamon sugar mixture evenly over the buttery surface.
Starting with the long side, roll up the dough slightly loosely, so it has room to rise. Cut dough into 12 1-inch thick slices using a sharp knife or string. Place slices cut-side-down in a greased 9x13 glass cake pan. Cover lightly with plastic wrap, and let rise until doubled in size, about a half hour.
Preheat oven to 375 degrees, bake rolls until tops are light and golden, about 20 minutes. Let cool in the pan while preparing the frosting.
Make the frosting: Using a hand mixer, whip together the butter, cream cheese, extracts, and salted caramel until thoroughly combined. Add in the powdered sugar, and then the milk slowly until the right consistency is reached.
Spread frosting over cooled cinnamon rolls, and drizzle with more caramel sauce.