Step 1: Preheat your oven to 400°F (200°C). Line a large baking sheet with parchment paper.
Step 2: Prepare the mango filling. In a medium bowl, combine the diced ripe mangoes, softened cream cheese, granulated sugar, fresh lime juice, and vanilla extract. Mix gently until just combined. Taste and adjust sugar if needed.
Step 3: Roll out the puff pastry. On a lightly floured surface, carefully unfold one sheet of thawed puff pastry. Roll it out slightly to smooth out creases and achieve an even thickness, to about a 10x10 inch square. Cut into four equal squares. Repeat with the second sheet of puff pastry, yielding eight squares in total.
Step 4: Fill the pastries. Place about 2-3 tablespoons of the mango cream cheese filling onto the center of each pastry square, leaving a generous border around the edges.
Step 5: Form the turnovers. Fold one corner of the pastry square over the filling to meet the opposite corner, forming a triangle. Gently press down on the edges to seal. Use a fork to crimp the edges firmly.
Step 6: Prepare the egg wash. In a small bowl, whisk together the beaten egg and 1 tablespoon of water.
Step 7: Egg wash and bake. Transfer the formed turnovers to the prepared baking sheet. Brush the tops of each turnover generously with the egg wash. Sprinkle with coarse sugar. Cut two or three small slits on the top of each turnover.
Step 8: Bake for 20-25 minutes, or until the pastries are puffed, golden brown, and the filling is bubbly. Transfer to a wire rack to cool slightly before serving.