Heat oven to 350°F. Line 13 x 9 x 2-inch baking pan with nonstick foil, leaving overhang.
In saucepan, melt chocolate and butter over medium heat, stirring until smooth. Remove from heat. Stir in sugar, salt and walnuts. Stir in eggs, one at a time. Stir in chips, flour and vanilla. Pour into prepared pan.
Bake in 350°F oven for 30 minutes.
In saucepan, melt butter and chocolate over medium heat, stirring until smooth. Stir in corn syrup, rum extract and salt. Keep warm.
Lift brownie from pan. While still warm, cut into 1-inch pieces. Place 4 pieces in dessert bowls. Top with 1/2 cup ice cream and 2 more pieces. Drizzle with sauce. Serve.