In a medium saucepan, bring brown sugar, butter, milk, cream, and salt to a boil over medium heat, stirring frequently; cook, stirring frequently, until an instant-read thermometer registers 230°F (110°C) to 235°F (113°C), 7 to 10 minutes. Remove from heat.

Pour caramel into the heatproof bowl of a stand mixer fitted with the whisk attachment. (See Note.) Slowly increase mixer speed to medium-low. Add vanilla, beating until combined. Gradually add confectioners’ sugar, beating until combined. Scrape bottom and sides of bowl. Beat at medium- high speed until smooth and well combined, 20 to 30 seconds. Use immediately while still warm.