To assemble the cake: Unmold the vanilla cake from the pan and place on a cutting board. Using a 6-inch springform cake pan ring, cut out 2 full circles from the cake. Wash the ring and place it in the center of a parchment paper-lined baking sheet. Fit a 32 inch-long sheet of parchment paper inside the ring to line it, and secure it closed with clear tape (the paper should stand 12 inches tall).
Break the cake scraps apart and place them inside the ring to completely cover the bottom, pressing to flatten them into an even layer. Using a pastry brush, brush 3⁄4 teaspoon of the lemon juice over the cake layer, and then spread half the pickled strawberry jam over the cake. Sprinkle with one-third of the milk crumbs, and press them into the jam to secure.
In a large bowl, whisk the cheesecake with the lemon curd until completely smooth. Spoon half of the lemon cheesecake over the milk crumbs and smooth the top. Place one of the full cake layers over the cheesecake layer and press gently to level. Brush the cake with the remaining 3⁄4 teaspoon lemon juice, and then repeat layering with the remaining pickled strawberry jam, half of the remaining milk crumbs, and the remaining lemon cheesecake, smoothing the top.
Place the remaining cake layer on top of the cheesecake and press gently to level. Spread the pickled strawberry frosting over the cake layer and smooth the top. Sprinkle with the remaining milk crumbs and then freeze the assembled cake for at least 12 hours to set the layers. Store the cake in the freezer until ready to serve, up to 2 weeks.
Remove the cake from the freezer, and remove the springform ring. Peel away and discard the parchment paper, and then transfer the cake to a serving plate or cake stand. Let the cake stand to defrost for at least 3 hours before slicing and serving.