In the bowl of a stand mixer, combine the milk, instant yeast, light brown sugar, eggs, butter, 3 ½ cups flour, salt and cinnamon. With a dough attachment, mix the ingredients until the flour is incorporated, about 4-5 minutes. If dough is too sticky, add flour 1 tablespoon at a time but it's better to be on the sticky side. Cover the dough with plastic wrap and let rest at room temperature for 1 hour or until it has doubled in size.
In a small bowl, mix together the brown sugar, cinnamon and salt for the filling. Set aside.
While dough is resting, make the apple filling. Heat butter in medium skillet over medium heat. Add apples, sugar, lemon juice, vanilla extract, cinnamon and salt. Cook for about 10 minutes or until apples are soft. Add cornstarch to thicken. Remove from heat and let cool.
Lightly dust your work surface with some flour. Remove dough from mixing bowl and punch down. Roll the dough into a rectangle about 24x16 inches. It doesn't have to be perfect. Spread the softened butter evenly over the dough. Sprinkle the cinnamon mixture over the butter and lightly push into the butter. Evenly sprinkle the apple pie mixture over the cinnamon mixture.
Starting with the long edge close to you, roll the dough into a log, trying to keep tight throughout. With a sharp knife, cut 12 rolls about 2 inches thick. Place the rolls in a 9x13 baking dish. Cover with plastic wrap and transfer to fridge to rise overnight. (You can let rise for 30 minutes at room temperature in a pinch)
Preheat oven to 350°F. Bake the rolls for 25-30 minutes or until golden brown on top and the middles are done.
While baking rolls, make the brown butter frosting. In a small saucepan, melt butter over medium heat and cook for about 5 minutes or until the butter is lightly browned and smells toasted. Remove from heat and let cool.
In a stand mixer, beat together cooled brown butter, cream cheese, powdered sugar, vanilla extract and salt until smooth. If the frosting is too thick, just add 1 tablespoon of milk at a time until it reaches your desired consistency.
Spread the frosting over the warmed rolls and serve immediately.