Preheat the oven to 375F. To make the filling, heat half of the berries with 1/4 cup water in a medium saucepan over medium-high heat. Cook for a few minutes, until the berries soften and burst.
In a small bowl, stir together the sugar and cornstarch. Stir into the berries along with the lemon juice and a pinch of salt. Bring to a boil and simmer for 1-2 minutes, until the mixture has thickened. Stir in the uncooked berries and remove from the heat. Set aside to cool slightly.
Divide the puff pastry in half and on a lightly floured surface, roll each piece into a 10 to 12-inch square, and cut each square into four squares. (Place the pastry squares on two parchment-lined baking sheets to avoid having to move them once filled.) Brush around the edge with some beaten egg and place a large spoonful of filling on each. Fold over and seal, making a triangle (or rectangle), and press down around the edges with the tines of a fork to seal.
Brush the tops with a bit of beaten egg and poke steam holes in the top with the fork or the tip of a sharp knife.
Bake for 20 minutes, or until deep golden. Makes 8 hand pies.