Preheat the oven to 350°F (175°C). Line the bottoms of three 9-inch cake pans with parchment paper and spray or grease the pans, then dust with flour (or use a nonstick spray with flour). If you have only two pans, you will bake two layers, then bake the third layer immediately after the first two finish.
In a large liquid measuring cup or medium bowl, whisk together the milk (1 1/2 cups), the 9 large egg whites, the vanilla extract (1 1/2 teaspoons), and the almond extract (1 teaspoon). Set this wet mixture aside.
In the bowl of an electric mixer fitted with the whisk attachment, combine the cake flour (3 1/3 cups), the granulated sugar for the cake (2 2/3 cups), the baking powder (2 tablespoons), and the salt (1 1/2 teaspoons). Mix on low speed just to combine.
With the mixer running on low, add the cubed, softened unsalted butter (18 tablespoons / 2 1/4 sticks) a few pieces at a time. Once all butter is added, increase the mixer speed to medium and beat until the mixture resembles moist crumbs and the largest pieces are about the size of small peas, about 3 minutes.
Pour half of the reserved milk-and-egg-white mixture into the mixer and beat on medium-high until the batter is light and fluffy, about 1 minute. Stop the mixer and scrape down the bowl.
With the mixer on low, stream in the remaining milk-and-egg-white mixture. Increase speed to medium-low and mix no more than 30 seconds. If the batter looks slightly lumpy, stop mixing; give a few gentle stirs by hand with a spatula—do not overmix.
Divide the batter evenly among the three prepared 9-inch pans and smooth the tops. Bake at 350°F until a toothpick inserted in the center comes out with a few crumbs but no wet batter, about 22–25 minutes.
Let the cakes rest in the pans for 5 minutes, then turn them out onto cooling racks. Cool completely before frosting, about 1½ hours.
While the cakes cool, make the raspberry filling. Add the raspberries (16 ounces, frozen then thawed or very ripe fresh) to a food processor and puree until smooth.
Place a fine-mesh sieve over a medium saucepan. Pour the raspberry puree into the sieve and press with the back of a spoon or spatula to strain out the seeds, collecting the seedless puree in the saucepan. Scrape the underside of the sieve to get as much puree as possible into the pan.
Whisk the cornstarch (2 1/2 tablespoons), the granulated sugar for the filling (2/3 cup), and the lemon juice (2 1/2 tablespoons) into the seedless puree until the cornstarch is mostly dissolved.
Heat the puree over medium-high, stirring constantly until it comes to a simmer. Reduce the heat to medium (or medium-low if your stove runs hot) and continue to cook, stirring continuously, until the mixture thickens to a quite-thick but spreadable consistency. Remove from heat and refrigerate until completely chilled; it will thicken further as it cools.
Make the white chocolate component: place the chopped white baking chocolate (12 ounces) and the heavy cream (1/4 cup) in a microwave-safe bowl. Microwave for 30 seconds, stir, then continue microwaving in 15-second intervals, stirring between intervals, until the chocolate is completely smooth. Let cool for about 5 minutes.
Make the white chocolate cream cheese frosting: in a large bowl, beat the cream cheese (28 ouncepkgs. cream cheese (16 ounces) at room temperature), the 1/2 cup unsalted butter (at room temperature), and 1/4 teaspoon salt at medium speed with a hand-held electric mixer until very creamy, 1–2 minutes.
Beat the cooled white chocolate and the vanilla extract (2 teaspoons) into the cream cheese mixture until smooth.
Gradually add the powdered sugar (7 cups), in batches, beating at low speed until blended after each addition. When all powdered sugar is incorporated, refrigerate the frosting about 30 minutes to help it firm up; it should be fairly thick to support the layers.
After chilling the frosting, spoon roughly 1 cup of frosting into a pastry bag to use for piping the dams around each cake layer. If you do not have a pastry bag, you can use a freezer bag with a corner snipped off. Spoon 3/4 cup of frosting into a small bowl and set aside to use as the crumb-coat frosting. Keep the remaining frosting chilled until assembly.
Level the cooled cake layers: use a large serrated knife to remove the domes from the tops of the cakes so each layer is flat.
Prepare your cake stand or serving platter: place four strips of parchment paper around the edges of the stand to form a square with an open middle (this keeps the stand clean). Place a small dollop of frosting in the center of the stand to anchor the first layer.
Place the first leveled cake layer on the stand, top side up. Pipe a ring (a dam) of the prepared frosting just inside the top edge of the cake to contain the filling.
Spread half of the chilled raspberry filling inside the piped ring, using an offset spatula to bring the filling to the edges of the dam. Spread 1 cup of the white chocolate frosting over the raspberry layer and smooth to the edges.
Place the second leveled cake layer on top, top side down. Repeat: pipe a ring of frosting just inside the top edge, spread the remaining raspberry filling to the edges of that ring, then spread 1 cup of white chocolate frosting over the raspberry layer.
Place the third leveled cake layer on top, top side down.
Apply the crumb coat: using the reserved 3/4 cup crumb-coat frosting, frost a thin, even layer over the top and sides of the assembled cake to seal in crumbs. Smooth as desired and remove any excess frosting. Refrigerate the cake for about 30 minutes, or until the crumb coat has set. Meanwhile, refrigerate the remaining frosting and then lightly rebeat it before final frosting if it has become too firm.
Finish frosting: frost the outside of the chilled, crumb-coated cake with the remaining white chocolate frosting as desired. Remove the parchment strips from around the base of the cake, and pipe a ring of frosting around the bottom edge if desired.
Garnish the top with additional raspberries if desired. Refrigerate the cake until ready to serve; bring to a suitable serving temperature before slicing.