Make the cake: Heat the oven to 350° F. Butter a 13-inch by 18-inch rimmed baking sheet. Line sheet with parchment and butter the parchment.
In a small bowl, whisk together the flour, baking powder, and salt. In a medium bowl, beat the butter, egg yolks, and ⅓ cup water on high until lightened, about 2 minutes. Gradually add ¼ cup sugar. Continue to beat until light and fluffy, about 3 minutes.
With an electric mixer, beat the egg whites in a large bowl until frothy. Gradually add the remaining ⅓ cup sugar. Increase speed to high and continue to beat until whites form stiff, glossy peaks, about 4 minutes.
Sift the flour mixture into the egg yolk mixture and stir to combine. Add 1 cup of the egg white mixture into the egg yolk mixture to loosen it, then gently fold in the remaining egg whites. Transfer the batter to the prepared baking sheet and smooth the top. Bake until the center of the cake bounces back when pressed gently, about 15 to 18 minutes.
Meanwhile, lay a clean kitchen towel on a work surface and dust generously with confectioners' sugar. As soon as the cake comes out of the oven, invert it onto the prepared towel. With the short side facing you, roll the warm cake and the towel into a cylinder. Transfer to a rack and let cool completely.
In a small bowl, combine the blackberries and granulated sugar. Set aside. In a medium bowl, combine the cream and confectioners’ sugar and, using and electric mixer, whip to stiff peaks.
To assemble, unroll the cooled cake. Spread the whipped cream evenly over the surface. Drizzle with the blackberries and their juices. Spread the berries and the cream together to combine, then roll the cake up again without the towel. Transfer cake to a plate, dust with confectioners sugar, slice, and serve.