Preheat the oven to 400°F (200°C) and line a baking sheet with parchment paper.
In a medium saucepan, combine the mixed berries, sugar, cornstarch, and lemon juice. Cook over medium heat until the mixture thickens and the berries have released their juices, about 5-7 minutes. Remove from heat and let cool.
Roll out the puff pastry on a lightly floured surface and cut into squares (about 4x4 inches).
Place a spoonful of berry mixture in the center of each square. Fold the pastry over the filling to form a triangle and press the edges to seal.
Brush the tops of the turnovers with the beaten egg and place them on the prepared baking sheet.
Bake for 20-25 minutes, or until golden brown and puffed. Let cool on a wire rack before serving.