In a medium saucepan, heat the butter and brown sugar together over medium heat, stirring, until the brown sugar has dissolved, 1 to 2 minutes. Bring the mixture to a boil, then reduce the heat to low and simmer, stirring continuously, for 2 minutes.
Add the milk and return to a boil, stirring continuously, until the mixture is well combined, 2 minutes.
Remove from the heat and cool to lukewarm, 20 minutes.
While whisking continuously, gradually add the confectioners’ sugar. Add additional milk 1 tablespoon at a time if the frosting becomes too stiff (it should be firm but still spreadable).