Preheat the oven to 400°F (200°C). Line one or two baking sheets with parchment paper and set aside.
Unfold the thawed puff pastry sheets on a lightly floured surface and cut each sheet into 9 equal squares.
In a mixing bowl, combine the chopped cooked chicken, whipped cream cheese, shredded Parmesan, and 4 to 6 teaspoons Sriracha, stirring until evenly mixed.
Place a portion of the chicken mixture down the center of each pastry square, oriented diagonally.
Fold two opposite corners of each square over the filling and press the edges to seal, forming a closed puff. Transfer to the prepared baking sheet.
Whisk the egg with 1 tablespoon water to make an egg wash. Brush each pastry lightly with the egg wash.
Bake for 15 to 18 minutes, or until the puffs are golden brown. Let cool slightly and serve warm or at room temperature.
