Melt butter in a heavy 3-quart saucepan over medium heat; stir in brown sugar and salt. Stir in corn syrup. Slowly pour in milk, stirring constantly until combined. Using a candy thermometer, cook to 248 degrees F (about 25 minutes), stirring occasionally (the mixture will be very bubbly). Remove saucepan from heat and stir in vanilla and nuts. Immediately pour into prepared pan.