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.