French Onion Chicken Pot Pie

Preheat oven to 425°F. Place chicken, 2 teaspoons of the olive oil, 1 teaspoon of the salt, and 1/2 teaspoon of the pepper in a large bowl; toss to coat. Place chicken, skin side up, on a baking sheet lined with aluminum foil. Bake in preheated oven until golden brown and a thermometer inserted into thickest portion of chicken registers 165°F, about 35 minutes. Remove chicken from oven, and let cool at room temperature until chicken is easy to handle, about 10 minutes. Remove and discard skin and bones. Using 2 forks, shred chicken into bite-size pieces; set aside. (You'll have about 3 cups.)

Unfold pastry sheet, and roll into a 9-inch square on a lightly floured work surface. Using a 3-inch round cookie cutter, cut pastry into 7 rounds. Discard excess dough. Place pastry rounds on a parchment paper-lined baking sheet. Brush tops evenly with egg. Bake in preheated oven until pastry is light brown and puffed, about 10 minutes. Remove from oven, and let cool on baking sheet.

While pastry rounds are cooling, heat butter and remaining 2 tablespoons olive oil in a 10-inch cast-iron skillet over medium. Add onions; cook, stirring often, until golden brown, 25 to 28 minutes, adjusting heat as needed to prevent burning. Stir in carrots and water; cook, stirring often, until onions are deep golden brown and carrots are tender-crisp, 5 to 6 minutes. Add garlic; cook, stirring constantly, until fragrant, about 1 minute. Add sherry, scraping up any browned bits from bottom of skillet using a wooden spoon. Cook, stirring often, until liquid is evaporated, about 45 seconds.

Stir in flour; cook, stirring constantly, until flour is light brown and onions become pasty, about 1 minute. Gradually stir in stock, and bring mixture to a simmer over medium. Simmer, stirring often, until mixture thickens, 3 to 5 minutes. Stir in reserved chicken, thyme, rosemary, and remaining 1 teaspoon salt and 1/2 teaspoon pepper. Remove from heat.

Nestle puff pastry rounds on top of chicken mixture in skillet. Sprinkle cheese evenly over chicken mixture and puff pastry rounds. Bake at 425°F until cheese is completely melted, 8 to 10 minutes. Remove from oven, and let cool slightly, about 5 minutes. Garnish with additional thyme leaves, and serve.