Preheat oven to 375°F. Grease a 13x9x2-inch baking pan; set aside. In a large bowl, beat the 1 cup butter with an electric mixer on medium to high speed for 30 seconds. Add the 1/2 cup brown sugar. Beat until combined, scraping side of bowl occasionally. Gradually add flour, beating on low speed just until combined (mixture will look slightly dry). Press mixture into prepared pan. Bake in the preheated oven for 10 minutes. Carefully spread preserves over hot crust.
In another large bowl, beat eggs on medium-high speed for 3 minutes. Add the 1 1/2 cups brown sugar, the cocoa powder, vanilla, and salt. Beat on low speed for 3 minutes. Gradually add the finely ground walnuts, beating just until combined.
Carefully spread chocolate mixture evenly over preserves. Bake about 30 minutes or until set. Cool completely in pan on a wire rack.
For the Icing: In a medium heavy saucepan, combine bittersweet chocolate and the 3 tablespoons butter. Cook and stir over low heat until melted. Stir in corn syrup and rum. Spread icing over cooled uncut cookies. If desired, top with the walnut halves. Let stand for 1 to 2 hours or until chocolate is set. Using a sharp knife, loosen edges of cookies from sides of pan. Cut into bars.