Instructions

1. Make the pickled strawberry jam: In a small pot, stir together the sugar, pectin, and salt. In a blender, purée the strawberries, then scrape the mixture into the pot. Stir in the sherry vinegar and white wine vinegar. Bring to a boil over high heat, then turn the heat to medium-low and simmer, stirring occasionally, until the jam is thickened and glossy, about 10 minutes. Remove from the heat and set aside to cool completely. (The jam will keep in an airtight container in the fridge for up to 2 weeks.)
2. Make the lemon curd: In a blender, process the sugar and lemon zest and juice until the sugar dissolves, about 30 seconds. Add the eggs and blend on low until smooth and foamy, about 15 seconds. Scrape the curd into a small pot. Rinse and dry the blender.
3. Cook the curd over low heat, stirring occasionally, until thickened and starting to boil, about 8 minutes. Meanwhile, in a small bowl, stir together the gelatin and 2 tablespoons of cold water and set aside to soften for 5 minutes. 
4. Pour the curd into the blender. Add the softened gelatin, butter, and salt and purée. Strain the curd through a fine-mesh strainer into a bowl, cover, and refrigerate until completely chilled, at least 1 hour. (The curd will keep in an airtight container in the fridge for up to 1 week.)
5. Make the milk crumbs: Position a rack in the center of the oven and preheat to 250°F. In a medium bowl, whisk together the flour, cornstarch, sugar, salt, and ½ cup of the dry milk. Using a silicone spatula, stir in the butter until the mixture forms small clusters. Spread the crumbs on a parchment-lined baking sheet and bake until dried and sandy, about 20 minutes. Transfer the crumbs to a rack and set aside to cool completely.
6. Transfer the crumbs to a medium bowl, breaking apart any that are larger than ½ inch in diameter, and toss with the remaining dry milk until evenly coated. Pour the white chocolate over the crumbs and toss with the spatula until evenly coated. Set aside, tossing every 5 minutes, until the chocolate is cooled completely and no longer sticky. (The crumbs will keep in an airtight container in the fridge for up to 1 week.)
7. Make the cheesecake: Position a rack in the center of the oven and preheat to 300°F. Line a 6-inch round cake pan with parchment. In a small bowl, whisk together the cornstarch and salt, then stir in the milk and egg until smooth. Set aside.
8. In a medium bowl, use a hand mixer on medium speed to beat the cream cheese until smooth and fluffy, about 2 minutes. Add the sugar and continue beating until fluffy, about 2 minutes more. Add the cornstarch slurry and beat until smooth and loose, about 3 minutes.
9. Scrape the batter into the prepared pan and bake until set at the edges but still loose in the center and not browned, about 20 minutes. Transfer the pan to a rack and set aside to cool completely. (The cheesecake will keep covered in the fridge for up to 1 week.)
10. Make the vanilla cake: Position a rack in the center of the oven and preheat to 350°F. Line an 8- by 12-inch baking sheet with parchment and grease with nonstick spray. In a stand mixer fitted with the paddle attachment, beat the sugar, butter, and brown sugar on medium-high until light and fluffy, about 3 minutes. Scrape down the sides of the bowl, add the eggs, and continue beating on medium-high until incorporated, about 3 minutes more. Turn the mixer speed to low and slowly drizzle in the buttermilk, oil, and vanilla. Turn the speed to medium-high and beat until the batter doubles in volume and turns white, about 6 minutes.
11. Turn the mixer speed to low, add the cake flour, baking powder, and salt and mix until just combined. Scrape the batter into the prepared baking sheet and smooth the top. Bake until a toothpick inserted into the center comes out clean, 28–30 minutes. Transfer the baking sheet to a rack and set aside to cool completely. (The cake will keep in the baking sheet, wrapped in plastic, in the fridge for up to 5 days.)
12. Make the frosting: In a small bowl, stir together the salt, citric acid, and ¼ cup plus 1 tablespoon of the pickled strawberry jam. In a stand mixer fitted with the paddle attachment, beat the butter and confectioners sugar on medium-high until pale and fluffy, about 3 minutes. Turn the speed to low, slowly pour in the jam mixture, and beat until smooth and fluffy, about 1 minute.
13. Assemble the cake: Unmold the vanilla cake and place on a cutting board. Using a 6-inch springform cake pan ring, cut out 2 full rounds from the cake. Wash and dry the ring, then place it in the center of a parchment-lined baking sheet. Fit a 32-inch-long sheet of parchment inside the ring to line it and secure it closed with clear tape (the parchment should stand 12 inches tall).
14. Break apart the cake scraps and place them inside the ring to completely cover the bottom, pressing to flatten them into an even layer. Using a pastry brush, brush ¾ teaspoon of the lemon juice over the cake layer, then spread half of the remaining pickled strawberry jam over the cake. Sprinkle with one-third of the milk crumbs and press them into the jam to secure.
15. 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 ¾ teaspoon of lemon juice, then repeat layering with the remaining pickled strawberry jam, half of the remaining milk crumbs, and the remaining lemon cheesecake, smoothing the top.
16. Place the remaining cake layer on top and press gently to level. Spread the frosting over the cake layer and smooth the top. Sprinkle with the remaining milk crumbs. 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.
17. Remove the springform ring from the cake and peel off and discard the parchment. Transfer the cake to a serving plate or cake stand. Set the cake aside to defrost for at least 3 hours before slicing and serving.