Soak candied fruit, orange zest, and lemon zest in spiced or dark rum 30 minutes.
Beat butter and next 4 ingredients at medium speed with an electric mixer 2 to 3 minutes or until creamy.
Add flour; beat until blended. Stir in fruit mixture. Divide dough into 4 portions, and shape each portion into an 8- x 2-inch log. Wrap in plastic wrap, and chill 1 hour.
Whisk together egg and 1 Tbsp. water. Unwrap logs, and brush with beaten egg. Sprinkle chopped walnuts over logs, pressing to adhere, and rewrap. Chill 30 minutes.
Preheat oven to 350°. Cut logs into 1/4-inch-thick slices; place 1 inch apart on 2 parchment paper-lined baking sheets. Bake 10 to 14 minutes or until edges are lightly browned, switching pans halfway through. Cool on baking sheets 5 minutes. Transfer to wire racks, and cool.