In a small saucepan over medium heat, combine the butter, brown sugar and milk. Whisk often until the butter is completely melted and the mixture begins to boil. Remove from heat and cool for 10 minutes.
Add the confectioners' sugar and vanilla. Whisk until smooth. Gently pour glaze in a circular motion over the cooled bundt cake and serve. Store cake covered at room temperature for up to 5 days.