Combine brown sugar, butter, and salt in a saucepan over medium heat. Stir until brown sugar is dissolved, about 3 minutes.
Add evaporated milk and continue stirring. Bring to a gentle boil and let bubble for about 4 minutes, stirring constantly to avoid sticking.
Remove from heat and allow to cool, about 5 minutes.
Mix confectioners sugar and vanilla extract into brown sugar mixture using an electric mixer until frosting caramelizes and thickens to the desired consistency.