Melt butter in a saucepan over medium heat. Add brown sugar and 2 tablespoons maple syrup; stir frequently until sugar is melted. Add milk and stir frequently it comes to a boil. Pour mixture into a mixing bowl and allow to cool slightly, 10 to 12 minutes.

Add powdered sugar and remaining 1 tablespoon maple syrup to the cooled mixture; beat with an electric mixer until smooth. Spread frosting while it's still warm for best results.