Preheat the oven to 400°. In a large bowl, mix the chicken, pear, shallot, chicken broth, tarragon, 1 teaspoon salt and 1/2 teaspoon pepper and set aside. In a small bowl, beat the egg with 1 teaspoon water; set aside.
Unfold and flatten the puff pastry sheet; cut into 4 squares and refrigerate 3 squares. On a lightly floured surface, roll out 1 pastry square into a 9-inch square. Pile one-quarter of the chicken mixture in the center. Lightly brush the 4 corners with the egg wash and pinch together to seal, leaving the seams open. Place the turnover on a baking sheet and refrigerate. Repeat with the remaining pastry squares, filling and egg wash.
Bake the turnovers until crisp and golden, about 25 minutes.