Grease bottom and sides of a 15" x 10" jellyroll pan; line bottom with wax paper. Grease and flour wax paper. Set aside.
Beat egg yolks at high speed with an electric mixer until foamy. Gradually add sugar, beating until thick and pale (about 2 minutes). Gradually add 1/4 cup cocoa, beating well. Stir in 1 teaspoon vanilla.
Beat egg whites at high speed until foamy. Add cream of tartar and salt; beat until stiff peaks form. Fold about one-fourth of beaten egg white into yolk mixture; gradually fold in remaining egg white. Sift cake flour over batter, and gently fold until combined.
Spread batter evenly into prepared pan. Bake at 325° for 16 to 17 minutes or until top springs back when lightly touched. Sift 1 to 2 tablespoons cocoa in a 15" x 10" rectangle on a cloth towel. When cake is done, immediately loosen from sides of pan, and turn out onto towel. Peel off wax paper. Starting at narrow end, roll up cake and towel together; cool on a wire rack, seam side down.
Pour sweetened condensed milk into a medium-size heavy saucepan. Cook over medium-low to medium heat, stirring constantly, 20 minutes or until milk turns a light caramel color. Remove from heat, and stir in 1 1/2 cups toasted coconut, chopped pecans, 2 teaspoons vanilla, and half-and-half.
Unroll cake, and remove towel. Spread cake with coconut filling. Carefully reroll cake without towel. Place cake, seam side down, on a wire rack over wax paper.
Stir together whipping cream and corn syrup in a saucepan; bring to a boil over medium heat. Remove from heat, and add chopped chocolate, stirring until smooth. Pour about three-fourths of chocolate mixture over cake, letting excess drip onto wax paper. Let remaining chocolate sit at room temperature until consistency of frosting (about 45 minutes). Spread remaining chocolate mixture onto ends of cake roll. Garnish, if desired. Let cake stand until chocolate is firm. Place cake on a serving platter.