In a saucepan, combine the butter, dark brown sugar and milk and bring to a boil over medium heat, stirring constantly. Reduce heat to medium-low and let the mixture bubble for 1-2 minutes, or until it turns an amber color (continue to stir constantly). Remove from the heat and stir in the vanilla. Let cool for 15 minutes.
In a large bowl, mix (using whisk attachments) the cooled butter mixture, powdered sugar and sea salt with a hand or stand mixer on medium speed until creamy, scraping down the sides of the bowl with a spatula if necessary. Turn mixer to medium speed and beat for 3 minutes.