Prepare the pan: Heat the oven to 180°C or 350°F. Grease and line a 23 x 33 cm baking pan with parchment paper.
Mix the batter: Whisk flour, sugar, baking powder, and salt. Beat in butter until sandy. Add lemon zest, eggs, milk, vanilla, and lemon juice, mixing until smooth. Fold in blackberries.
Bake: Spread the batter in the pan and bake for 30 to 35 minutes. Cool completely.
Make the frosting: Beat butter until creamy. Add powdered sugar, milk, and vanilla. Beat in cooled melted white chocolate until smooth.
Assemble: Cut three six-inch cake rounds. Layer cake, frosting, and blackberry jam twice. Finish with frosting, fresh blackberries, and lemon zest.
Chill: Refrigerate the cake for at least two hours before slicing and serving.