In the bowl of a stand mixer fitted with the dough hook, combine the milk, yeast, and brown sugar. Let sit 5-10 minutes, until bubbly on top. Add the butter, eggs, 3 1/2 - 4 cups flour, vanilla, and salt. Using the dough hook, mix until the flour is completely incorporated, about 4-5 minutes. If the dough is still sticky, add an additional 1/4 cup flour, adding more as needed until the dough is smooth to touch.Cover the bowl with plastic wrap and let sit at room temperature for 1 hour or until doubled in size.Meanwhile, mix the filling. In a bowl, combine the brown sugar, sugar, and cinnamon. Butter a 9×13 inch baking dish.Punch the dough down and roll out onto a lightly floured surface, creating a large rectangle about 12 x 18 inches. Spread the butter evenly over the dough. Sprinkle on the cinnamon sugar. Starting with the long edge closest to you, roll the dough into a log, keeping it tight as you go. When you reach the edge, pinch along the edge to seal. Using a sharp knife, cut into 12-15 rolls. Place the rolls into the prepared baking dish. Cover with plastic wrap and let rise 15-20 minutes.Preheat the oven to 350° F. Bake the rolls for 25 to 30 minutes, or until golden brown.Meanwhile, make the icing. Add the butter to a pot set over medium heat. Allow the butter to brown lightly until it smells toasted, about 2-3 minutes. Remove from the heat. In a bowl, beat the cream cheese until fluffy. Add the brown butter, beating to combine. Then, add the powdered sugar, caramel, vanilla, and a pinch of salt. Beat until creamy. Spread the icing over the warm rolls. Serve and enjoy!