Scatter blackberries on a plate covered in paper towel to thaw and drain. Preheat oven to 180°C. Grease and line base and sides of a 30cm x 20cm lamington tin with baking paper, allowing paper to overhang long sides.
Put butter and sugar in the small bowl of an electric mixer and beat until pale and creamy. Add yolks and beat until combined. Sift flours and baking powder then add to butter mixture in three batches with motor running until mixture forms a sticky dough.
Turn dough onto a clean surface. Bring together and knead with a little extra gluten-free flour, if sticky, until just smooth. Press mixture evenly over base of tin. Use a straight-sided glass or bottle to roll over dough to smooth out surface. Chill for 15 minutes.
Prick base with a fork then bake for 15 minutes or until
lightly golden. Cool slightly. Spread jam over base.
Put eggwhites in a large bowl and beat with a whisk to loosen. Stir in coconut and sugar until combined. Sprinkle coconut mixture evenly over jam. Sprinkle blackberries on top.
Bake for 25-30 minutes or until coconut is golden. Allow to cool in tin for 30 minutes before lifting and sliding onto a wire rack to cool. Transfer to a board to cut in half lengthways then into long slices or squares to serve.