For the dough: Combine the flour, sugar and salt in a bowl. Cut the butter into small pieces and with a pastry blender, cut the butter into the flour mixture until it resembles coarse crumbs. Add the egg yolk. Quickly knead until the dough comes together. Wrap in plastic wrap and refrigerate 30 minutes.
For the filling: Drain and puree the lychees.
In a bowl, stir together the jam sugar, butter and quark until well combined. Stir in the eggs, pureed lychees and rum, finally stir in the custard powder.
Preheat the oven to 200°C (approximately 400°F). Line the bottom of a springform pan with parchment paper and grease the sides of the pan.
Press 2/3 of the dough into the bottom of the pan. With the remaining dough, form a high edge, pressing the side into the bottom to seal. Pour the filling into the pan and bake on the bottom rack of the oven until set, about 1 hour, covering with foil during the final 15 minutes of baking. Cool in the pan on a wire rack, then refrigerate until well chilled before serving.