For the sweetened whipped cream: In the bowl of a stand mixer fitted with the paddle attachment, add the cream cheese and mix until smooth; transfer to a bowl.
In the same mixer bowl, combine the heavy whipping cream and sugar. Switch to the whisk attachment and whip on high speed until soft peaks form. Add the cream cheese back to the bowl and continue whisking on high until the cream forms stiff peaks.
For the berry filling: Combine the sugar, cornstarch and salt in a large saucepan and stir to combine. Add the water, lemon juice and lemon zest; stir to combine. Turn the heat to medium-low and add the blueberries and raspberries. Cook, stirring occasionally, until the mixture begins to bubble and thicken, about 5 minutes. Cook an additional minute, then add the strawberries, butter and vanilla. Gently stir until the butter has melted, then transfer to a bowl and refrigerate until completely chilled, about 2 hours.
For the jaconde: Line two half-sheet pans with silicone baking mats. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment, combine 1 cup of the confectioners' sugar and the 8 tablespoons butter and mix until smooth. Gradually add the 1/2 cup egg whites and mix to combine. Gradually add the cake flour and mix until a paste is formed. Divide the paste between bowls based on how many colors are desired. Add a drop or two of food coloring to each of the bowls and stir to achieve the desired colors.
Transfer the paste to a pastry bag fitted with a small open tip and pipe your chosen pattern on the baking mats. Freeze the patterns until frozen solid. Once frozen solid, leave the decor in the freezer and make the cake batter.
Preheat the oven to 425 degrees F.
In a small bowl, whisk together the almond flour and all-purpose flour. Set aside.
Place a stand-mixer bowl over a pot of simmering water to build a double boiler. Add the 3 whole eggs and 1/4 cup of the granulated sugar and whisk vigorously until the sugar has dissolved. Transfer the bowl to the stand mixer and beat on high speed with the whisk attachment until the bowl is cool to the touch and the mixture ribbons. Transfer the mixture to a large bowl and set aside. Wash and dry the bowl and whisk very well. (Note: A bowl or whisk with any reside will not beat egg whites properly.)
In the clean mixer bowl, combine the 2 egg whites and salt. Beat with the clean whisk attachment on high speed until frothy. Slowly add the remaining 1/4 cup granulated sugar. Continue whisking until the egg whites hold soft peaks; set aside.
Sift the almond flour mixture over the whole egg mixture; using a silicone spatula, gently fold to incorporate. Fold in the egg whites, followed by the melted butter.
Remove the prepared pans with the frozen decor from the freezer. Immediately divide the cake batter between the two pans and gently spread into an even layer.
Bake until the edges are barely brown and the cake is springy, about 5 minutes.
Allow to cool, then run a small knife along the edges of the pans to release the cakes. Sift the remaining 1/2 cup confectioners' sugar over the cakes.
Place a piece of parchment over each cake, followed by an inverted baking sheet. Flip over so the cakes land decorated-side up on the baking sheets. Gently release the baking sheets and carefully peel away the baking mats from the cakes. If needed, use a paring knife to help with this process.
Place one Flaky Pastry Layer on a cardboard cake round. Fill a pastry bag with half the whipped cream; place the rest back in the refrigerator. Cut the tip of the bag to create a 1/2-inch opening. Pipe a band of whipped cream along the perimeter of the pastry disk and spread one-third of the berry filling inside.
Place a second pastry layer on top, pipe the whipped cream along the perimeter again, and add another third of the berry filling. Top with a third pastry layer and repeat with the whipped cream and the remaining berry filling. Top with the final pastry layer, then wrap the cake gently in plastic, and freeze for 30 minutes or until the whipped cream feels set. Refrigerate the remaining whipped cream in the piping bag.
Remove the layered cake from the freezer and pipe whipped cream to coat the sides with a crumb coat. Refrigerate for 20 minutes to set.
Measure the height and circumference of the layered cakes and trim the jaconde to fit around the sides and top. (Note: You will likely need to cover the entire cake in multiple strips.) Gently press the jaconde to the sides and top of the cake.
Place the remaining half of the whipped cream in a pastry bag fitted with a star tip and pipe rosettes on top of the cake.
Add the flour and salt to a large bowl and stir to combine. Add the butter and toss to combine. With your fingers, begin to "smush" the butter into flat pieces. Continue smushing until all the butter is flattened but none of the pieces are larger than a quarter. The flour will look like coarse cornmeal.
Add the water, quickly folding it in with a wooden spoon to distribute the moisture. (Do not stir vigorously--the dough should appear shaggy.) Use your hands to break up any large clumps of moisture. Then use your knuckles or a palm to smear the dough up the sides of the bowl (a technique known as fraisage). This distributes the butter over a larger surface to achieve larger flakes and compact the dough. Once you've gone around the bowl one time, fold the dough over itself a few times to pick up any dry bits from the bottom. Gently press the dough into a disk. (Note: The dough should remain very cool to the touch throughout the process. If it approaches room temperature, refrigerate it for 5 to 10 minutes to chill.)
Transfer the dough to a lightly floured work surface and roll out to a 10-by-12-inch rectangle. Perform a letter fold by folding one of the short ends towards the middle and the other short end over the first, like folding a business letter.
Turn the dough 90 degrees, so the long side faces you, and perform a second letter fold. Turn the dough 90 degrees again; perform a third letter fold.
If the dough still feels cool, it can be used immediately. Otherwise, wrap it in plastic wrap and chill for at least 20 minutes. (The dough can be refrigerated for up to 1 week or frozen for up to 1 month.)
Preheat the oven to 375 degrees F. Line a half-sheet pan with parchment paper.
Divide the dough into four even pieces. Roll each piece into a rough 10-inch round. Place the rounds onto the prepared sheet pan, placing a piece of parchment in between each round to separate. Cover with plastic wrap and refrigerate for 20 minutes.
Trim each round to a perfect 10-inch circle; dock each one. Bake the rounds on individual sheet pans, placing a piece of parchment over the rounds and then placing a second sheet pan on the parchment to "tame" the puff. (Note: Don't press the pan; just let it rest.) Bake for 15 minutes with the second sheet pan on top. Remove the second sheet pan and the top piece of parchment and bake until uniformly golden brown, 10 to 15 minutes more. If you are baking in batches, refrigerate the unbaked rounds. Allow the baked rounds to cool completely.
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