Preheat the oven to 400 degrees F and line two baking sheets with parchment paper.

In a medium bowl, beat the cream cheese with an electric mixer on high speed until smooth. Add the sugar, lemon juice, lemon zest, and vanilla; beat until smooth then reserve.

On a lightly floured work surface, roll out each sheet of thawed puff pastry to a 10-inch square. Cut each square into 4 equal squares and transfer to the prepared baking sheets, leaving 2 inches between each square. Prick the pastry squares with a fork. Spoon 1 heaping tablespoon of the cream cheese mixture into the center of each square, spreading in a small circle about 1 1/2 inches in diameter. Spoon 1 tablespoon of the jam on top of the cream cheese mixture and top with 4 blackberries.

Use a sharp knife to make four diagonal cuts from the corners of each pastry square to within 1 inch of the center. Fold every other corner up over the filling to the center, overlapping the points and pinching to seal.