In a small saucepan, bring the frozen berries and sugar to a boil over medium heat for about 5 minutes.
Transfer the berries to a bowl with a spoon, leaving the berry juice in the pot.
Mix the starch with the water until smooth. Add the red juice to the pot with the berry juice and bring to a boil.
Stir in the starch mixture once it boils, then remove from the heat. Add the berries back into the pot and mix with the thickened juice.
Preheat the oven to 180°C (top/bottom heat) or 160°C (fan oven). Grease a loaf pan and dust it with semolina to prevent sticking.
For the cake, cream together the soft butter, sugar, and vanilla sugar in a large bowl. Beat in the eggs, one at a time. Mix in the quark until fully combined.
In a separate bowl, whisk together the pudding powder, flour, semolina, baking powder, and salt. Gradually add the dry ingredients to the quark mixture, mixing until smooth.
Wash the lemon thoroughly in hot water, dry it, and grate the zest. Stir about 1 teaspoon of lemon zest into the quark mixture for added flavor.
Pour half of the cake mixture into the prepared loaf pan, then add a layer of the berry mixture. Top with the remaining mixture.
Bake the cheesecake for about 60–70 minutes, or until a skewer inserted in the center comes out clean. Let the cheesecake cool completely in the pan before removing. Slice and enjoy!