Preheat oven to 400°F with rack in lower third of oven. Melt butter in a deep 10-inch ovenproof skillet over medium-high. Add leek and carrots. Cook, stirring often, until softened, about 6 minutes. Sprinkle with flour; cook, stirring constantly, 1 minute. Stir in broth; let mixture come to a simmer. Simmer, stirring constantly, until mixture thickens, 1 to 2 minutes. Stir in shredded chicken, peas, cream, thyme, salt, and pepper. Remove from heat; let cool 10 minutes.

Whisk together egg and 1 tablespoon water in a small bowl. Roll pastry sheet into a 12-inch square on a lightly floured surface. Cut into 16 (3-inch) squares. Arrange squares on top of chicken mixture in skillet, brushing each square with egg mixture and slightly overlapping squares to cover surface of chicken mixture. Place skillet on a rimmed baking sheet.

Transfer baking sheet with skillet to preheated oven. Bake until top is browned and filling is bubbly, about 30 minutes. Let stand 10 minutes, and serve.