Preheat oven to 325F. Line a 9-inch springform pan with parchment on the bottom. Beat cream cheese and sugar until completely smooth, about 3 minutes, scraping the bowl twice. Add eggs one at a time, beating just until combined. Mix in sour cream, vanilla, and ginger. Pour into prepared pan and bake 35-40 minutes until edges are set but center still jiggles slightly. Cool completely, then refrigerate at least 4 hours or overnight. Once firm, release from pan and keep chilled until assembly.

Preheat oven to 350F. Grease two 9-inch round cake pans, line with parchment rounds, and grease the parchment. Whisk together flour, baking soda, ginger, cinnamon, and cloves in a medium bowl until well combined.

In a large bowl, whisk melted butter, molasses, and sugar until smooth. Add eggs one at a time, whisking well after each. Stir in buttermilk. Add dry ingredients in three additions, stirring just until no streaks of flour remain. Divide evenly between prepared pans, about 3 cups batter each.

Bake 28-32 minutes until cakes spring back when lightly touched and a toothpick inserted in the center comes out clean. Cool in pans 15 minutes, then turn out onto wire racks to cool completely. Peel off parchment.

Beat cream cheese and butter together until completely smooth and fluffy, about 4 minutes, scraping the bowl frequently. Gradually add powdered sugar on low speed, then increase to medium and beat until light and fluffy, about 2 minutes. Beat in vanilla. If too thick, add heavy cream 1 teaspoon at a time.

Place one gingerbread cake layer on a serving plate. Spread a thin layer of frosting on top. Carefully place the chilled cheesecake layer in the center. Spread another thin layer of frosting on top of cheesecake. Place second gingerbread layer on top, flat side up. Apply a thin crumb coat of frosting all over the cake, then refrigerate 30 minutes. Finish with remaining frosting, smoothing sides and top. Refrigerate until ready to serve.