Crust: In a food processor, combine flour and sugar. Add butter and pulse until it resembles coarse meal. Add 3 Tbsp of whipping cream or milk and pulse until mixture forms a dough. If mixture is still too dry, add remaining cream or milk. Do not overprocess the dough. Remove from food processor and divide into 4 equal parts. Press the quarters of dough onto the bottom and sides of four 4 1/2 inch loose bottom tart tins. Place in freezer for half an hour.
Heat oven to 375 F or 190°C Prick bottoms of tart shells. Line tart shells with tin foil and fill with baking weights or beans. Bake in preheated oven for 10 minutes. Remove weights and tin foil and return to oven for 10 minutes or until crust is golden. Turn oven up to 400 F or 200 CLet cool slightly on a rack while preparing the filling.
Filling: In a medium bowl combine Quark, egg yolks, sugar, lemon zest, vanilla and salt until smooth. Divide the mixture between the four tart shells (it's alright if they are still warm). Sprinkle 1/4 cup of blackberries over each tart, spreading them out as necessary. Return tarts to oven and bake for 20-30 minutes, or until the filling is golden and set. Remove from oven and let cool on rack in tins for 10 minutes. Remove tart tins and let tarts cool fully.
Blackberry Glaze: In a small saucepan, combine water and berries. Simmer over medium low heat, stirring and mashing berries, until berries are soft and falling apart, about 5 minutes. Strain water and berry mixture, squeezing as much juice as possible from the berries. You should have about 1/3 cup of juice. If you have less, add a little water to make 1/3 cup. Return juice to saucepan. Stir together sugar and cornstarch in a small bowl and add to juice. Bring mixture back to a simmer briefly until it thickens. Spoon glaze over tarts, spreading it to the edges with the back of a spoon. Refrigerate tarts until glaze is cool.