Grease a square springform pan (24 x 24 cm; alternatively round, 26 cm Ø) and dust with flour. Separate 1 egg for the bottom. Knead 175 g flour, 50 g sugar, vanilla sugar, 1 pinch of salt, egg yolk and 125 g butter in pieces with the dough hooks of the mixer to a smooth dough.
Cover and chill for about 30 minutes.
For the cheese mixture, wash the lime hot and dab dry. Finely grate the rind. Squeeze the lime. Cream 75 g butter and 130 g sugar with the whisk of the mixer. Gradually add 3 eggs, the remaining egg white and cream cheese.
Stir in lime zest, juice and starch. Preheat the oven (electric cooker: 175 °C/circulating air: 150 °C/gas: see manufacturer).
Press the dough in the form to a base. Prick several times with a fork. Spread 200 g jam on the dough. Spread the cheese mixture with a tablespoon on the jam and smooth it down. Bake in a hot oven for about 1 hour.
Remove from the oven and let it cool down for about 10 minutes. Carefully remove from the springform pan. Let it cool down.
Sort the blackberries and wash if necessary. Wash the currants and remove from the panicles. Mix with the rest of the jam. Cut the cake into pieces of about 5 cm. Spread fruits on top.