Lightly coat a 15 x 10–inch jelly-roll pan with cooking spray; line bottom with parchment paper. Coat paper with cooking spray; dust with breadcrumbs. Set aside.
Sift together cocoa, flour, baking powder, baking soda, and salt into a medium bowl. Set aside.
Place egg yolks in a medium mixing bowl; beat with a mixer at high speed 2 minutes. Gradually add 1/4 cup granulated sugar and vanilla, beating until thick and pale (about 2 minutes). Using clean, dry beaters, beat egg whites in a medium bowl at high speed until foamy. Gradually add remaining 1/2 cup granulated sugar, 1 tablespoon at a time, beating until stiff peaks form.
Gently stir one-third of egg white mixture into yolk mixture; gently fold in one-third of cocoa mixture. Repeat procedure twice.
Pour batter into prepared pan, spreading evenly. Bake at 375° for 13 minutes or until cake springs back when touched lightly in center. Loosen the cake from sides of pan, and turn out onto a dish towel dusted with 2 tablespoons of powdered sugar. Carefully peel off parchment paper, and cool for 1 minute. Starting at narrow end, roll up cake and towel together. Place, seam side down, on a wire rack, and cool.
Unroll cake carefully, and remove towel. Combine 2 cups whipped topping, 1/4 teaspoon ground cinnamon, ground ginger, and ground cloves in a bowl, stirring well. Spread whipped topping mixture evenly over cake, leaving a 1/2-inch border around outside edges. Reroll cake, and place seam side down on a platter. Cover and chill for 1 hour. Dust cake with the remaining 1 tablespoon powdered sugar.