In a medium bowl beat butter with an electric mixer on medium to high speed for 30 seconds.
Gradually add powdered sugar, beating well.
Beat in bourbon or milk and vanilla. If necessary, beat in enough milk, 1 teaspoon at a time, to make frosting spreading consistency. Set aside until ready to use.
For the Brownies:
Preheat oven to 350° F. Grease an 8 x 8 x 2-inch baking pan; set aside.
In a medium saucepan, combine sugar, butter, and the water. Cook and stir over medium heat just until boiling. Remove from heat.
Stir in 1 cup chocolate pieces until melted. Add eggs and vanilla, beating with a wooden spoon just until combined.
Stir in flour, baking soda, and salt. Stir in chopped walnuts. Pour batter into the prepared baking pan, spreading evenly.
Bake for 20 to 25 minutes or until a wooden toothpick inserted near the center comes out clean and edges start to pull away from sides of pan.
Place pan on a wire rack. Brush top of hot brownies with bourbon. Cool completely.
Spread uncut brownies with Bourbon Frosting. Cut into bars. Drizzle top with melted chocolate.