Preheat the oven to 350 degrees F (180 degrees C). Lightly grease a 9x13-inch baking dish. Slice rolls in half horizontally, keeping tops and bottoms connected.
Stir together brown sugar, cinnamon, 8 tablespoons butter, and ⅛ teaspoon salt in a small bowl until well combined and set aside. Spread bottom half of buns with ¾ of butter mixture and place top halves on top. Place rolls in the prepared baking dish; spread remaining butter mixture on top.
Bake in the preheated oven for 20 minutes. Let stand for 5 minutes before icing.
Meanwhile, beat remaining 1 tablespoon butter with cream cheese, remaining salt, vanilla, and confectioner’s sugar with an electric mixer until smooth. Spread icing over warm rolls.
