Preheat oven to 350 degrees. Lightly coat an 8-inch springform cake pan with cooking spray. In a large bowl, whisk together cocoa powder, flour, sugar, baking soda, baking powder, and salt. Whisk in egg yolk, buttermilk, 1/4 cup warm water, oil, and vanilla. Pour batter into pan and bake until cake pulls away from sides and a toothpick inserted in center comes out clean, about 15 minutes. Let cool completely in pan on a wire rack, 1 1/2 hours. Cut one or two pieces parchment paper 1 inch taller than pan side and place between cake and inside of pan, overlapping if necessary (parchment should fit snugly around entire circumference of cake).
Place blackberries in a medium bowl and lightly mash with a fork. Stir in vanilla ice cream. Working quickly, spread ice cream mixture in an even layer over cooled cake with a small offset or rubber spatula. Freeze until ice cream is very firm, 2 hours. In a small bowl, stir sorbet until spreadable. Spread in an even layer over ice cream and freeze until set, 1 hour (or, covered, up to 1 week). To serve, unmold cake and peel away parchment.