Preheat the oven to 350 degrees F (175 degrees C). Grease and flour two 9-inch round cake pans and line the bottoms with parchment paper.
Whisk flour, cocoa, baking powder, baking soda, cinnamon, salt, and allspice together in a small bowl for the cake.
Combine sugar, oil, and applesauce in a large bowl with an electric mixer on low speed. Add pumpkin and vanilla extract. Beat in eggs on medium speed, 1 at a time, mixing well after each addition. Add flour mixture gradually until combined; finish mixing with a rubber spatula.
Pour batter into the prepared cake pans. Sprinkle chopped semisweet chocolate over the top of each. Swirl chocolate into the batter using a metal spatula.
Bake in the preheated oven until a toothpick inserted into the center of each cake comes out clean, 25 to 30 minutes. Cool in pans for 5 minutes. Loosen the sides, invert, remove parchment paper, and let cool on wire racks, about 30 minutes.
Meanwhile, combine cream cheese, powdered sugar, and vanilla extract for the filling in a mixing bowl. Beat using an electric mixer, increasing speed, until creamy. Remove 2/3 of the mixture and set aside. Add cocoa to remaining mixture; beat until creamy.
Beat heavy cream in another bowl on medium-high speed until stiff peaks form. Fold 2/3 of the whipped cream into the white cream cheese mixture. Fold remaining 1/3 of whipped cream into the chocolate cream cheese mixture.
Cut each cooled cake layer in half horizontally. Spread 1/2 of the white cream cheese mixture onto 1 layer of cake, to about 1/4-inch from the edge. Sprinkle 1/3 chopped semisweet chocolate over filling, gently pressing down with fingers. Add next layer; spread with chocolate cream cheese mixture and sprinkle with 1/3 of chopped semisweet chocolate; press down. Add third cake layer, spread with remaining white cream cheese. Sprinkle with remaining chocolate, pressing gently. Top with last layer.
Place bittersweet chocolate in the top of a double boiler over simmering water. Stir frequently, scraping down the sides with a rubber spatula to avoid scorching, until chocolate is melted, about 5 minutes. Remove from heat, add in powdered sugar and butter. Whisk until smooth. Pour over the top of the cake. Spread glaze evenly across the top and on the sides.
If you test the chocolate mixture for taste do not be tempted to add sugar as it may not taste very sweet. Your opinion will change once the cake is assembled!
There are two types of pumpkin lovers: subtle and spicy. If you are of the subtle variety like me, this recipe is perfect the way it is. If you prefer to really taste the spices (cinnamon, nutmeg, allspice, etc.) you will probably want to add more.
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