Preheat oven to 350°.
Coat 2 (9-inch) round cake pans with cooking spray, and dust with 1 tablespoon flour.
Beat the sugar and shortening at medium speed of a mixer 5 minutes or until well-blended. Add egg white and egg; beat well. Combine cocoa and food coloring in a small bowl; stir well with a whisk. Add to sugar mixture; mix well.
Lightly spoon 2 1/4 cups flour into dry measuring cups; level with a knife. Combine with salt. Add flour mixture to sugar mixture alternately with buttermilk, beginning and ending with flour mixture. Add vanilla; mix well. Combine vinegar and baking soda in a small bowl; add to batter, mixing well.
Pour batter into prepared pans. Sharply tap pans once on counter to remove air bubbles. Bake at 350° for 28 minutes or until a wooden pick inserted in center comes out clean. Cool in pans 10 minutes; remove from pans. Cool completely on wire racks.
Place 1 cake layer on a plate; spread with 1/3 cup Cooked Buttercream Frosting, and top with another cake layer. Spread the remaining frosting over the top and sides of the cake. Store cake loosely covered in refrigerator.