Make the cake: Heat oven to 350°. Grease and flour two 9″ cake pans, and set aside. Whisk together flour, baking powder, and salt in a bowl; set aside. Whisk together milk and vanilla in a bowl; set aside. In a small bowl, whisk together cinnamon, allspice, and cloves; set spices aside. In the bowl of a stand mixer fitted with a paddle, cream shortening and sugar on medium-high speed until evenly combined, about 2 minutes. Add egg whites one at a time, beating well after each addition. On low speed, alternately add dry ingredients in 3 batches and wet ingredients in 2 batches. Increase speed to high and beat until batter is smooth, about 5 seconds. Pour half the batter into one prepared pan. Add spices to remaining batter in bowl; stir until evenly combined. Pour spiced batter into second prepared pan; smooth tops of both batters with a rubber spatula. Drop pans lightly on a counter to expel any large air bubbles. Bake cakes until a toothpick inserted in middle comes out clean, about 30 minutes. Let cakes cool for 20 minutes in pans; invert onto wire racks, and let cool. Using a serrated knife, halve each cake horizontally to produce four layers; set aside.
Make the icing and assemble the cake: Bring sugar, milk, butter, and cocoa to a boil in a 6-qt. copper saucepan or Dutch oven over medium-high heat; attach a candy thermometer to side of pan, and cook, stirring bottom constantly with a wooden spoon, until thermometer reads 232°, about 30 minutes. Remove from heat and immediately stir in vanilla and baking soda; let sit for 1 minute. Using a hand mixer, beat chocolate mixture on medium-low speed until the consistency of loose pudding, about 1 minute. Immediately place one spiced cake layer on a cake stand and pour 1⁄2 cup chocolate icing over; top with one white cake and pour 1⁄2 cup icing over. Place second spiced cake over icing and pour 1⁄2 cup icing over; top with second white cake, and pour remaining icing around top edge of cake and then over the center. Quickly spread icing over sides of cakes; press walnuts into sides of cake until sides are covered. Let icing cool completely before serving. Store at room temperature.