Prepare the oven and pan: Heat the oven to 180°C, or 160°C for a fan oven. Grease and line a 23 x 33 cm baking pan.
Mix the dry ingredients: Whisk together the flour, sugar, baking powder, and salt. Add the softened butter and mix until fine crumbs form.
Finish the batter: Add lemon zest, eggs, milk, and vanilla. Beat until smooth, then mix in the lemon juice. Fold in the chopped blackberries.
Bake: Spread the batter in the pan and bake for 30 to 35 minutes, until a skewer inserted in the center comes out clean. Cool completely.
Make the frosting: Beat the butter until pale, then gradually add powdered sugar and milk. Beat in the melted, slightly cooled white chocolate.
Assemble: Mix the soak ingredients. Cut the cake into three round layers. Brush each layer lightly with the soak, then add frosting and blackberry jam between the layers.
Chill and decorate: Chill the assembled cake for at least 1 hour. Finish with fresh blackberries and lemon zest.