Coat a 3 1/2- to 4-quart slow cooker with cooking spray; set aside.
Whisk the flour, granulated sugar, 2/3 cup of the cocoa powder, baking powder, salt, and espresso powder together in a large bowl. Add the egg, egg yolk, milk, oil, and vanilla and stir until combined. Pour the batter into the slow cooker. Sprinkle the remaining 1/3 cup cocoa powder and the brown sugar over the top of the batter. Scatter the chopped chocolate evenly over the top. Pour in the water, but do not stir.
Cover and cook on the LOW setting until the cake pulls away from the edge and the center of the cake wobbles slightly, 1 hour 30 minutes to 2 hours. You will see some bubbling around the edges. Spoon into serving dishes, dust with powdered sugar, and serve with fresh berries and vanilla ice cream.