Preheat oven to 400°F with rack in lower third position. On a lightly floured surface, roll puff pastry into a 10-inch square. Cut 2 to 3 (2-inch) slits horizontally across the center of pastry, spacing each 1-inch apart. Place on a large plate or small baking sheet, chill while you build the filling.
Melt butter in a deep 10-inch ovenproof skillet over medium. Add shallots and garlic; cook, stirring often, until softened, about 2 minutes. Stir in wine and bring to a boil over medium; cook until reduced by half, about 2 minutes, stirring occasionally.
Whisk together flour and chicken broth in a measuring cup; stir into skillet and cook, whisking constantly, until mixture thickens, 1 to 2 minutes. Stir in chicken, lemon juice, capers, parsley, ½ teaspoon salt, and ¼ teaspoon pepper. Remove from heat.
Whisk together egg and water in a small bowl. Place pastry square over hot chicken mixture, letting ends extend over sides of skillet. Brush pastry evenly with egg wash. Sprinkle with remaining ⅛ teaspoon each salt and pepper. Brush the undersides of the puff pastry corners with the egg wash; press to the skillet. Place skillet on a rimmed baking sheet.
Bake until top is browned and filling is bubbly, about 20 minutes. Remove from oven and let stand 10 minutes. Sprinkle with fresh parsley and serve.
