1. Cake: Preheat oven to 350 degrees. Brush two 8-inch round cake pans with butter. Line pans with parchment rounds; brush parchment

2. Whisk together flour, baking powder, salt, cinnamon, ginger, and nutmeg until thoroughly combined, 30 seconds. Beat eggs with granulated sugar on medium-high speed until pale, fluffy, and doubled in volume, about 5 minutes. While still beating, slowly drizzle in butter, then beat in vanilla until combined. Reduce speed to low and add all but 1/2 cup of flour mixture, beating just until combined. Toss carrots with remaining flour mixture to evenly coat, then fold into batter. Divide batter evenly between pans; smooth tops with an offset spatula.

3. Bake until cakes are puffed slightly and a tester inserted into centers comes out clean, 33 to 37 minutes. Transfer pans to a wire rack; let cool 10 minutes. Turn cakes out onto rack; remove parchment and let cool completely.

4. Frosting: Meanwhile, beat cream cheese on medium-high speed until smooth and creamy, about 2 minutes. While still beating, gradually add chocolate until smooth, scraping down bowl as needed. Add butter and lemon juice; beat to combine. Reduce speed to low; beat in confectioners' sugar, 1/4 cup at a time, until smooth and combined.

5. Cut each cake horizontally into 2 layers with a serrated knife. Place bottom layer of one cake, cut-side up, on a cake stand, preferably revolving, or a plate lined with parchment strips. Spread evenly with 3/4 cup frosting. Top with other half of cake, cut-side down; spread evenly with another 3/4 cup frosting. Repeat layering and frosting with second cake. Spread 1 cup frosting evenly over top and sides to create a crumb coat. Refrigerate, uncovered, until frosting is firm, at least 30 minutes.

6. Transfer remaining frosting to a pastry bag fitted with a petal tip, such as Ateco or Wilton #104. Starting at top center of cake and holding tip vertically, pipe frosting while slowly turning stand or plate, gradually working outward toward edges to create a swirl effect. Repeat process down side of cake, starting at top edge. (Sides will be easier to decorate if you keep large end of tip touching or almost touching cake, holding pointed tip perpendicular to cake while tracing down side.) If frosting starts to get too loose, refrigerate briefly just until firm, but not long enough to harden. Cake can be stored, uncovered, at room temperature up to 12 hours or refrigerated, uncovered, up to 2 days; bring to room temperature before serving.