For the cake:Unsalted butter, for greasing the pan
1 1/2 cups (6 3/4 ounces) all-purpose flour
1 1/2 teaspoons baking powder
1/2 teaspoon kosher salt
4 eggs, separated
3/4 cup sugar, divided
6 ounces blackberries, plus more for serving
1/3 cup whole milk
Seeds from 1 vanilla bean
For the milk-soak:
3/4 cup whole milk
3/4 cup heavy cream
3/4 cup sweetened condensed milk
3/4 cup evaporated milk
1 teaspoon vanilla extract Preheat the oven to 350° F and butter an 8- by 8-inch baking dish. In a small bowl, whisk together the flour, baking powder, and salt. In the bowl of a stand mixer, or working with an electric hand mixer, beat together the egg yolks and 1/2 cup of sugar on medium speed until very pale and thick, 3 to 4 minutes. Meanwhile, halve the berries lengthwise. Now, beat in 1/3 cup of milk and the vanilla bean seeds. Add the flour mixture and mix until just combined. In a large bowl with clean beaters (transfer the batter to another bowl if necessary), whip the egg whites for about 1 minute, until foamy and white. With the mixer running, slowly add the remaining 1/4 cup sugar; continue to beat until you have shiny, medium-stiff peaks, about 2 minutes. Stir about 1/4 of the egg white mixture into the batter, to loosen it up a bit, then use a rubber spatula to fold the remaining white into the batter. Transfer to prepared pan and place the halved berries on top. Bake until golden brown and a toothpick inserted comes out clean, 35 to 40 minutes. Let cool for 5 minutes while you mix together the milk mixture. In a small bowl, stir together all the ingredients for the milk soak. Use a butter knife to cut around the edges of the cake. Then use a toothpick or a fork to make holes all over the top of the cake. pour about 1 cup of the milk mixture all over the cake and let it soak in. Then pour another cup over the cake. Cover and chill for at least 4 hours or up to overnight. Serve the cake with remaining mixture and more blackberries. Leftovers will keep, well-wrapped, in the fridge for up to 2 days.