Line the bottom of a springform pan (26 cm Ø) with baking paper. Separate the eggs. Beat the egg whites, 2-3 tbsp. cold water and 1 pinch of salt until stiff, while adding sugar and 1 packet of vanilla sugar.
Beat the egg yolks separately. Sift flour, starch and baking powder on top and fold in. Brush into the form. Bake in the preheated oven (electric oven: 175 °C/circulating air: 150 °C/gas: level 2) for 25-30 minutes.
Let it cool down.
Wash the limes and grate them dry. Grate the peel of 1 1/2-2 limes. Then halve these 2 limes and squeeze out the juice (approx. 7 tbsp.).
Soak gelatine in cold water. Wash the mint and put something aside for decoration. Cut the rest into strips. Mix yoghurt, rum, 75 g brown sugar, lime zest and juice. Stir in mint.
Squeeze the gelatine, dissolve at low heat. Stir in 3-4 tablespoons of yoghurt cream and then stir into the remaining cream. Chill for approx. 25 minutes until it begins to gel.
Cut the cake base 1 x horizontally. Peel 2 limes so that the white skin is completely removed. Cut into thin slices and dab dry a little.
Beat 250 g cream until stiff, fold into the yoghurt cream. Close a cake ring around the lower cake base. Spread the yoghurt cream on top. Place the second cake base on top and chill for about 4 hours.
Whip 500 g cream until stiff, pouring in 2 sachets of vanilla sugar. Spread the cake with it. Press the lime slices onto the edge of the cake. Decorate the cake with 3 tsp. brown sugar and mint.