In a small saucepan, bring brown sugar, ⅓ cup water, and salt to a boil over medium-high heat, stirring constantly, until sugar is dissolved. Remove from heat and stir in ½ cup butter until melted. Let cool completely.

In a large bowl, beat cooled sugar mixture, cream cheese, and remaining ½ cup butter with a mixer at medium speed until smooth and creamy. Reduce speed to low, and gradually add confectioners’ sugar, beating until smooth. If frosting is too thick to spread, beat in milk, 1 tablespoon at a time, to achieve a spreadable consistency. Use immediately.