How to Make It
Preheat oven to 350°F. Grease 3 (8-inch) round cake pans with shortening; dust with flour. Whisk together flour, sugar, baking powder, cinnamon, nutmeg, and salt in a large bowl. Whisk together egg yolks, 2/3 cup water, oil, lemon juice, and vanilla in a separate bowl until smooth. Whisk egg yolk mixture into flour mixture until batter is just combined.
Beat egg whites in a medium bowl with an electric mixer fitted with the whisk attachment on medium-high speed until foamy, about 1 minute. Add cream of tartar; beat until soft peaks form, 1 to 2 minutes. Fold one-third of egg white mixture into batter. Add remaining egg white mixture; gently fold until just combined. Divide batter evenly among prepared pans.
Bake in preheated oven until cakes are light golden brown and a wooden pick inserted in cake centers comes out clean, 16 to 20 minutes. Cool cakes in pans on a wire rack 10 minutes. Remove cakes from cake pans; transfer to wire racks, and let cool completely, about 30 minutes.
When ready to assemble cakes, make the ginger–cream cheese frosting: Place grated ginger in center of a square of folded cheesecloth; gather edges of cheesecloth, and squeeze ginger over a small bowl to equal 2 tablespoons juice. Discard solids; set aside ginger juice.
Beat cream cheese, sugar, vanilla, and salt in a medium bowl with an electric mixer fitted with the whisk attachment on medium-high speed until light and fluffy, about 3 minutes, stopping to scrape down sides of bowl as needed.
Stir together ginger juice and heavy cream in a small bowl. With mixer running on medium-high speed, gradually drizzle ginger juice mixture into cream cheese mixture; beat until stiff peaks form, 2 to 3 minutes. Use immediately.
Place 1 cake layer on a serving platter. Spread top with about 1/3 cup frosting. Spoon 3/4 cup frosting into a ziplock plastic bag; cut a 1/3-inch hole in one corner. Pipe a 1/3-inch-tall ring of frosting just inside the top edge of frosted cake layer (to create a barrier). Spread half of the cooled compote (about 2/3 cup) in an even layer inside piped ring. Top with second cake layer; repeat process with frosting, frosting ring, and remaining compote. Top with third cake layer; spread remaining frosting over top and sides of cake. Refrigerate assembled cake at least 4 hours or up to 1 day. Serve chilled.